771 lines
20 KiB
Plaintext
771 lines
20 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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\end_header
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\begin_body
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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
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\begin{equation}
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\troprp qp^{-1}=\troprp pq=\troprp qp^{\dagger}\label{eq:regular translation unitarity}
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\end{equation}
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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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Similarly, the full regular operators can be composed
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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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better wording
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\end_layout
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\end_inset
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,
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\begin_inset Formula
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\begin{equation}
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\troprp ac=\troprp ab\troprp bc\label{eq:regular translation composition}
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\end{equation}
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\end_inset
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but truncation breaks this,
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\begin_inset Formula $\truncated{\troprp ac}L\ne\truncated{\troprp ab}L\truncated{\troprp bc}L.$
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\end_inset
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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 Subsection
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Multiple-scattering problem
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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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\rcoeffp p=\rcoeffincp p+\sum_{q\in\mathcal{P}\backslash\left\{ p\right\} }\tropsp pq\outcoeffp q\label{eq:particle total incident field coefficient a}
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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 (many scatterers)
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\end_layout
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\begin_layout Standard
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For a system of many scatterers, Kristensson
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\begin_inset CommandInset citation
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LatexCommand cite
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after "sect. 9.2.2"
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key "kristensson_scattering_2016"
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literal "false"
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\end_inset
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derives only the extinction cross section formula.
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Let us re-derive it together with the many-particle scattering and absorption
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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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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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\[
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\vect E\left(\vect r\right)=\sum_{\tau,l,m}\left(\rcoeffptlm{\square}{\tau}lm\vswfrtlm{\tau}lm\left(\vect r-\vect r_{\square}\right)+\outcoeffptlm{\square}{\tau}lm\vswfouttlm{\tau}lm\left(\vect r-\vect r_{\square}\right)\right),
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\]
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\end_inset
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where
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||
\begin_inset Formula $\rcoeffp{\square},\outcoeffp{\square}$
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||
\end_inset
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||
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||
are the vectors of VSWF expansion coefficients of the incident and total
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||
scattered fields, respectively, at origin
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||
\begin_inset Formula $\vect r_{\square}$
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\end_inset
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||
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.
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In principle, one could evaluate
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||
\begin_inset Formula $\outcoeffp{\square}$
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\end_inset
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||
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using the translation operators (REF!!!) and use the single-scatterer formulae
|
||
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||
\begin_inset CommandInset ref
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||
LatexCommand eqref
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||
reference "eq:extincion CS single"
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||
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||
\end_inset
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||
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–
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||
\begin_inset CommandInset ref
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||
LatexCommand eqref
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reference "eq:absorption CS single"
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||
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||
\end_inset
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||
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||
with
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||
\begin_inset Formula $\rcoeffp{}=\rcoeffp{\square},\outcoeffp{}=\outcoeffp{\square}$
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||
\end_inset
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||
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||
to obtain the cross sections.
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||
However, this is not suitable for numerical evaluation with truncation
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||
in multipole degree; hence we need to express them in terms of particle-wise
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||
expansions
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\begin_inset Formula $\rcoeffp p,\outcoeffp p$
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||
\end_inset
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||
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.
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The original incident field re-expanded around
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||
\begin_inset Formula $p$
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||
\end_inset
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||
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||
-th particle reads according to
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||
\begin_inset CommandInset ref
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||
LatexCommand ref
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||
reference "eq:regular vswf translation"
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||
plural "false"
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||
caps "false"
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||
noprefix "false"
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||
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||
\end_inset
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||
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||
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||
\begin_inset Formula
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||
\begin{equation}
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||
\rcoeffincp p=\troprp p{\square}\rcoeffp{\square}\label{eq:a_inc local from global}
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||
\end{equation}
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||
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||
\end_inset
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||
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||
whereas the contributions of fields scattered from each particle expanded
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||
around the global origin
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||
\begin_inset Formula $\vect r_{\square}$
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||
\end_inset
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||
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||
is, according to
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||
\begin_inset CommandInset ref
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||
LatexCommand ref
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||
reference "eq:singular vswf translation"
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||
plural "false"
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||
caps "false"
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||
noprefix "false"
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||
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||
\end_inset
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||
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||
,
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||
\begin_inset Formula
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||
\begin{equation}
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||
\outcoeffp{\square}=\sum_{q\in\mathcal{P}}\troprp{\square}q\outcoeffp q.\label{eq:f global from local}
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||
\end{equation}
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||
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||
\end_inset
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||
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||
Using the unitarity
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||
\begin_inset CommandInset ref
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||
LatexCommand ref
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||
reference "eq:regular translation unitarity"
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||
plural "false"
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||
caps "false"
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||
noprefix "false"
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||
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||
\end_inset
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||
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||
and composition
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||
\begin_inset CommandInset ref
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||
LatexCommand ref
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||
reference "eq:regular translation composition"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
properties, one has
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
\rcoeffp{\square}^{\dagger}\outcoeffp{\square} & =\rcoeffincp p^{\dagger}\troprp p{\square}\troprp{\square}q\outcoeffp q=\rcoeffincp p^{\dagger}\sum_{q\in\mathcal{P}}\troprp pqf_{q}\nonumber \\
|
||
& =\sum_{q\in\mathcal{P}}\left(\troprp qp\rcoeffincp p\right)^{\dagger}f_{q}=\sum_{q\in\mathcal{P}}\rcoeffincp q^{\dagger}f_{q},\label{eq:atf form multiparticle}
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
where only the last expression is suitable for numerical evaluation with
|
||
truncated matrices, because the previous ones contain a translation operator
|
||
right next to an incident field coefficient vector (see Sec.
|
||
TODO).
|
||
Similarly,
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
\left\Vert \outcoeffp{\square}\right\Vert ^{2} & =\outcoeffp{\square}^{\dagger}\outcoeffp{\square}=\sum_{p\in\mathcal{P}}\left(\troprp{\square}p\outcoeffp p\right)^{\dagger}\sum_{q\in\mathcal{P}}\troprp{\square}q\outcoeffp q\nonumber \\
|
||
& =\sum_{p\in\mathcal{P}}\sum_{q\in\mathcal{P}}\outcoeffp p^{\dagger}\troprp pq\outcoeffp q.\label{eq:f squared form multiparticle}
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
Substituting
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:atf form multiparticle"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:f squared form multiparticle"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
into
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:scattering CS single"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:absorption CS single"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
, we get the many-particle expressions for extinction, scattering and absorption
|
||
cross sections suitable for numerical evaluation:
|
||
\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\sum_{p\in\mathcal{P}}\rcoeffincp p^{\dagger}\outcoeffp p=-\frac{1}{2k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\Re\sum_{p\in\mathcal{P}}\rcoeffincp p^{\dagger}\left(\Tp p+\Tp p^{\dagger}\right)\rcoeffp p,\label{eq:extincion CS multi}\\
|
||
\sigma_{\mathrm{scat}}\left(\uvec k\right) & = & \frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\sum_{p\in\mathcal{P}}\sum_{q\in\mathcal{P}}\outcoeffp p^{\dagger}\troprp pq\outcoeffp q\nonumber \\
|
||
& & =\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\sum_{p\in\mathcal{P}}\sum_{q\in\mathcal{P}}\rcoeffp p^{\dagger}\Tp p^{\dagger}\troprp pq\Tp q\rcoeffp q,\label{eq:scattering CS multi}\\
|
||
\sigma_{\mathrm{abs}}\left(\uvec k\right) & = & -\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\sum_{p\in\mathcal{P}}\Re\left(\outcoeffp p^{\dagger}\left(\rcoeffincp p+\sum_{q\in\mathcal{P}}\troprp pq\outcoeffp q\right)\right)\nonumber \\
|
||
& & =\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\sum_{p\in\mathcal{P}}\mbox{TODO}.\label{eq:absorption CS multi}
|
||
\end{eqnarray}
|
||
|
||
\end_inset
|
||
|
||
An alternative approach to derive the absorption cross section is via a
|
||
power transport argument.
|
||
Note the direct proportionality between absorption cross section
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:absorption CS single"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
and net radiated power for single scatterer
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:Power transport"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $\sigma_{\mathrm{abs}}=-\eta_{0}\eta P/2\left\Vert \vect E_{0}\right\Vert ^{2}$
|
||
\end_inset
|
||
|
||
.
|
||
In the many-particle setup (with non-lossy background medium, so that only
|
||
the particles absorb), the total absorbed power is equal to the sum of
|
||
absorbed powers on each particle,
|
||
\begin_inset Formula $-P=\sum_{p\in\mathcal{P}}-P_{p}$
|
||
\end_inset
|
||
|
||
.
|
||
Using the power transport formula
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:Power transport"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
particle-wise gives
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\sigma_{\mathrm{abs}}\left(\uvec k\right)=-\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\sum_{p\in\mathcal{P}}\left(\Re\left(\rcoeffp p^{\dagger}\outcoeffp p\right)+\left\Vert \outcoeffp p\right\Vert ^{2}\right)\label{eq:absorption CS multi alternative}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
which seems different from
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:absorption CS multi"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
, but using
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:particle total incident field coefficient a"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
, we can rewrite it as
|
||
\begin_inset Formula
|
||
\begin{align*}
|
||
\sigma_{\mathrm{abs}}\left(\uvec k\right) & =-\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\sum_{p\in\mathcal{P}}\Re\left(\outcoeffp p^{\dagger}\left(\rcoeffp p+\outcoeffp p\right)\right)\\
|
||
& =-\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\sum_{p\in\mathcal{P}}\Re\left(\outcoeffp p^{\dagger}\left(\rcoeffincp p+\sum_{q\in\mathcal{P}\backslash\left\{ p\right\} }\tropsp pq\outcoeffp q+\outcoeffp p\right)\right).
|
||
\end{align*}
|
||
|
||
\end_inset
|
||
|
||
It is easy to show that all the terms of
|
||
\begin_inset Formula $\sum_{p\in\mathcal{P}}\sum_{q\in\mathcal{P}\backslash\left\{ p\right\} }\outcoeffp p^{\dagger}\tropsp pq\outcoeffp q$
|
||
\end_inset
|
||
|
||
containing the singular spherical Bessel functions
|
||
\begin_inset Formula $y_{l}$
|
||
\end_inset
|
||
|
||
are imaginary,
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
TODO better formulation
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
so that actually
|
||
\begin_inset Formula $\sum_{p\in\mathcal{P}}\Re\left(\sum_{q\in\mathcal{P}\backslash\left\{ p\right\} }\outcoeffp p^{\dagger}\tropsp pq\outcoeffp q+\outcoeffp p^{\dagger}\outcoeffp p\right)=\sum_{p\in\mathcal{P}}\Re\left(\sum_{q\in\mathcal{P}\backslash\left\{ p\right\} }\outcoeffp p^{\dagger}\troprp pq\outcoeffp q+\outcoeffp p^{\dagger}\outcoeffp p\right)=\sum_{p\in\mathcal{P}}\Re\left(\sum_{q\in\mathcal{P}\backslash\left\{ p\right\} }\outcoeffp p^{\dagger}\troprp pq\outcoeffp q+\outcoeffp p^{\dagger}\troprp pp\outcoeffp p\right)=\sum_{p\in\mathcal{P}}\sum_{q\in\mathcal{P}}\outcoeffp p^{\dagger}\troprp pq\outcoeffp q,$
|
||
\end_inset
|
||
|
||
proving that the expressions in
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:absorption CS multi"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:absorption CS multi alternative"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
are equal.
|
||
\end_layout
|
||
|
||
\end_body
|
||
\end_document
|