Merge finite-cs.lyx into finite.lyx
Former-commit-id: c42c118767e8fde87df5655946c3489a70627033
This commit is contained in:
Marek Nečada
parent
3aa4de7e77
commit
64e122b937
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@ -660,6 +660,27 @@ My implementation.
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Maybe put the numerical results separately in the end.
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\end_layout
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\begin_layout Section
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TODOs
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\end_layout
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\begin_layout Itemize
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Consistent notation of balls.
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How is the difference between two cocentric balls called?
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\end_layout
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\begin_layout Itemize
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Abstract.
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\end_layout
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\begin_layout Itemize
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Example results.
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\end_layout
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\begin_layout Itemize
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Concrete comparison with other methods.
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\end_layout
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\begin_layout Standard
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\begin_inset CommandInset include
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LatexCommand include
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@ -680,6 +701,10 @@ literal "true"
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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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\begin_inset CommandInset include
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LatexCommand include
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filename "finite-old.lyx"
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@ -690,12 +715,6 @@ literal "true"
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\end_layout
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\begin_layout Standard
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\begin_inset CommandInset include
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LatexCommand include
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filename "finite-cs.lyx"
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literal "true"
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\end_inset
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@ -713,6 +732,10 @@ literal "true"
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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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\begin_inset CommandInset include
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LatexCommand include
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filename "infinite-old.lyx"
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@ -721,6 +744,11 @@ literal "true"
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\end_inset
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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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@ -1,770 +0,0 @@
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#LyX 2.4 created this file. For more info see https://www.lyx.org/
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\lyxformat 583
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\begin_document
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\begin_header
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\save_transient_properties true
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\origin unavailable
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\textclass article
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\use_default_options true
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\maintain_unincluded_children false
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\language english
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\language_package default
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\inputencoding utf8
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\fontencoding auto
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\font_roman "default" "TeX Gyre Pagella"
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\font_sans "default" "default"
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\font_typewriter "default" "default"
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\font_math "auto" "auto"
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\font_default_family default
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\use_non_tex_fonts false
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\font_sc false
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\font_roman_osf true
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\font_sans_osf false
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\font_typewriter_osf false
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\font_sf_scale 100 100
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\font_tt_scale 100 100
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\use_microtype false
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\use_dash_ligatures false
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\graphics default
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\default_output_format default
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\output_sync 0
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\bibtex_command default
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\index_command default
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\float_placement class
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\float_alignment class
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\paperfontsize default
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\spacing single
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\use_hyperref true
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\pdf_title "Sähköpajan päiväkirja"
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\pdf_author "Marek Nečada"
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\pdf_bookmarks true
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\pdf_bookmarksnumbered false
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\pdf_bookmarksopen false
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\pdf_bookmarksopenlevel 1
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\pdf_breaklinks false
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\pdf_pdfborder false
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\pdf_colorlinks false
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\pdf_backref false
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\pdf_pdfusetitle true
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\papersize default
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\use_geometry false
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\use_package amsmath 1
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\use_package amssymb 1
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\use_package cancel 1
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\use_package esint 1
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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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\use_package stmaryrd 1
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\use_package undertilde 1
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\cite_engine basic
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\cite_engine_type default
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\biblio_style plain
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\use_bibtopic false
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\use_indices false
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\paperorientation portrait
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\suppress_date false
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\justification true
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\use_refstyle 1
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\use_minted 0
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\use_lineno 0
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\index Index
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\shortcut idx
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\color #008000
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\end_index
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\secnumdepth 3
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\tocdepth 3
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\paragraph_separation indent
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\paragraph_indentation default
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\is_math_indent 0
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\math_numbering_side default
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\quotes_style english
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\dynamic_quotes 0
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\papercolumns 1
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\papersides 1
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\paperpagestyle default
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\tablestyle default
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\tracking_changes false
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\output_changes false
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\html_math_output 0
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\html_css_as_file 0
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\html_be_strict false
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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 $\openball R{\vect r_{\square}}$
|
||||
\end_inset
|
||||
|
||||
, we can describe the EM fields as if there was only a single scatterer,
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\vect E\left(\vect r\right)=\sum_{\tau,l,m}\left(\rcoeffptlm{\square}{\tau}lm\vswfrtlm{\tau}lm\left(\vect r-\vect r_{\square}\right)+\outcoeffptlm{\square}{\tau}lm\vswfouttlm{\tau}lm\left(\vect r-\vect r_{\square}\right)\right),
|
||||
\]
|
||||
|
||||
\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
|
||||
|
||||
with
|
||||
\begin_inset Formula $\rcoeffp{}=\rcoeffp{\square},\outcoeffp{}=\outcoeffp{\square}$
|
||||
\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
|
||||
|
||||
.
|
||||
The original incident field re-expanded around
|
||||
\begin_inset Formula $p$
|
||||
\end_inset
|
||||
|
||||
-th particle reads according to
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
reference "eq:regular vswf translation"
|
||||
plural "false"
|
||||
caps "false"
|
||||
noprefix "false"
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\rcoeffincp p=\troprp p{\square}\rcoeffp{\square}\label{eq:a_inc local from global}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
whereas the contributions of fields scattered from each particle expanded
|
||||
around the global origin
|
||||
\begin_inset Formula $\vect r_{\square}$
|
||||
\end_inset
|
||||
|
||||
is, according to
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
reference "eq:singular vswf translation"
|
||||
plural "false"
|
||||
caps "false"
|
||||
noprefix "false"
|
||||
|
||||
\end_inset
|
||||
|
||||
,
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\outcoeffp{\square}=\sum_{q\in\mathcal{P}}\troprp{\square}q\outcoeffp q.\label{eq:f global from local}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
Using the unitarity
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
reference "eq:regular translation unitarity"
|
||||
plural "false"
|
||||
caps "false"
|
||||
noprefix "false"
|
||||
|
||||
\end_inset
|
||||
|
||||
and composition
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
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
|
||||
|
|
@ -144,7 +144,7 @@ employing point group symmetries and decomposing the problem to decrease
|
|||
\end_deeper
|
||||
\end_deeper
|
||||
\begin_layout Subsection
|
||||
Motivation
|
||||
Motivation/intro
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
|
@ -162,9 +162,9 @@ The basic idea of MSTMM is quite simple: the driving electromagnetic field
|
|||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
However, the expressions appearing in the re-expansions are fairly complicated,
|
||||
and the implementation of MSTMM is extremely error-prone also due to the
|
||||
various conventions used in the literature.
|
||||
The expressions appearing in the re-expansions are fairly complicated, and
|
||||
the implementation of MSTMM is extremely error-prone also due to the various
|
||||
conventions used in the literature.
|
||||
Therefore although we do not re-derive from scratch the expressions that
|
||||
can be found elsewhere in literature, we always state them explicitly in
|
||||
our convention.
|
||||
|
|
@ -326,8 +326,8 @@ vector spherical harmonics
|
|||
|
||||
\begin_inset Formula
|
||||
\begin{align*}
|
||||
\vsh 1lm & =\\
|
||||
\vsh 2lm & =\\
|
||||
\vsh 1lm & =TODO\\
|
||||
\vsh 2lm & =fixme\\
|
||||
\vsh 3lm & =
|
||||
\end{align*}
|
||||
|
||||
|
|
@ -452,16 +452,16 @@ noprefix "false"
|
|||
\end_inset
|
||||
|
||||
inside a ball
|
||||
\begin_inset Formula $\openball 0R$
|
||||
\begin_inset Formula $\openball 0{R^{>}}$
|
||||
\end_inset
|
||||
|
||||
with radius
|
||||
\begin_inset Formula $R$
|
||||
\begin_inset Formula $R^{>}$
|
||||
\end_inset
|
||||
|
||||
and center in the origin; however, if the equation is not guaranteed to
|
||||
hold inside a smaller ball
|
||||
\begin_inset Formula $B_{0}\left(R_{<}\right)$
|
||||
\begin_inset Formula $B_{0}\left(R\right)$
|
||||
\end_inset
|
||||
|
||||
around the origin (typically due to presence of a scatterer), one has to
|
||||
|
|
@ -470,7 +470,7 @@ noprefix "false"
|
|||
\end_inset
|
||||
|
||||
to have a complete basis of the solutions in the volume
|
||||
\begin_inset Formula $\openball 0R\backslash B_{0}\left(R_{<}\right)$
|
||||
\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
|
|
@ -492,11 +492,11 @@ The single-particle scattering problem at frequency
|
|||
\end_inset
|
||||
|
||||
can be posed as follows: Let a scatterer be enclosed inside the ball
|
||||
\begin_inset Formula $B_{0}\left(R_{<}\right)$
|
||||
\begin_inset Formula $B_{0}\left(R\right)$
|
||||
\end_inset
|
||||
|
||||
and let the whole volume
|
||||
\begin_inset Formula $\openball 0R\backslash B_{0}\left(R_{<}\right)$
|
||||
\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
|
||||
\end_inset
|
||||
|
||||
be filled with a homogeneous isotropic medium with wave number
|
||||
|
|
@ -598,6 +598,19 @@ noprefix "false"
|
|||
\end_inset
|
||||
|
||||
.
|
||||
The outgoing VSWF expansion coefficients
|
||||
\begin_inset Formula $\outcoefftlm{\tau}lm$
|
||||
\end_inset
|
||||
|
||||
are related to the induced electric (
|
||||
\begin_inset Formula $\tau=1$
|
||||
\end_inset
|
||||
|
||||
) and magnetic (
|
||||
\begin_inset Formula $\tau=2$
|
||||
\end_inset
|
||||
|
||||
) multipole polarisation amplitudes of the scatterer.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
|
@ -780,7 +793,7 @@ literal "true"
|
|||
|
||||
.
|
||||
Let the field in
|
||||
\begin_inset Formula $\openball 0R\backslash B_{0}\left(R_{<}\right)$
|
||||
\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
|
||||
\end_inset
|
||||
|
||||
have expansion as in
|
||||
|
|
@ -795,11 +808,11 @@ noprefix "false"
|
|||
|
||||
.
|
||||
Then the net power transported from
|
||||
\begin_inset Formula $B_{0}\left(R_{<}\right)$
|
||||
\begin_inset Formula $B_{0}\left(R\right)$
|
||||
\end_inset
|
||||
|
||||
to
|
||||
\begin_inset Formula $\openball 0R\backslash B_{0}\left(R_{<}\right)$
|
||||
\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
|
||||
\end_inset
|
||||
|
||||
via by electromagnetic radiation is
|
||||
|
|
@ -811,7 +824,7 @@ P=\frac{1}{2k^{2}\eta_{0}\eta}\left(\Re\left(\rcoeffp{}^{\dagger}\outcoeffp{}\ri
|
|||
\end_inset
|
||||
|
||||
In realistic scattering setups, power is transferred by radiation into
|
||||
\begin_inset Formula $B_{0}\left(R_{<}\right)$
|
||||
\begin_inset Formula $B_{0}\left(R\right)$
|
||||
\end_inset
|
||||
|
||||
and absorbed by the enclosed scatterer, so
|
||||
|
|
@ -897,7 +910,7 @@ usual
|
|||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Subsection
|
||||
\begin_layout Subsubsection
|
||||
Cross-sections (single-particle)
|
||||
\end_layout
|
||||
|
||||
|
|
@ -957,8 +970,792 @@ reference "eq:plane wave expansion"
|
|||
Multiple scattering
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
If the system consists of multiple scatterers, the EM fields around each
|
||||
one can be expanded in analogous way.
|
||||
Let
|
||||
\begin_inset Formula $\mathcal{P}$
|
||||
\end_inset
|
||||
|
||||
be an index set labeling the scatterers.
|
||||
We enclose each scatterer in a ball
|
||||
\begin_inset Formula $B_{\vect r_{p}}\left(R_{p}\right)$
|
||||
\end_inset
|
||||
|
||||
such that the balls do not touch,
|
||||
\begin_inset Formula $B_{\vect r_{p}}\left(R_{p}\right)\cap B_{\vect r_{q}}\left(R_{q}\right)=\emptyset;p,q\in\mathcal{P}$
|
||||
\end_inset
|
||||
|
||||
,
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
TODO bacha, musejí být uzavřené!
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
so there is a non-empty volume
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
jaksetometuje?
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Formula $\openball{\vect r_{p}}{R_{p}^{>}}\backslash B_{\vect r_{p}}\left(R_{p}\right)$
|
||||
\end_inset
|
||||
|
||||
around each one that contains only the background medium without any scatterers.
|
||||
Then the EM field inside each such volume can be expanded in a way similar
|
||||
to
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
reference "eq:E field expansion"
|
||||
plural "false"
|
||||
caps "false"
|
||||
noprefix "false"
|
||||
|
||||
\end_inset
|
||||
|
||||
, using VSWFs with origins shifted to the centre of the volume:
|
||||
\begin_inset Formula
|
||||
\begin{align}
|
||||
\vect E\left(\omega,\vect r\right) & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\left(\rcoeffptlm p{\tau}lm\vswfrtlm{\tau}lm\left(k\left(\vect r-\vect r_{p}\right)\right)+\outcoeffptlm p{\tau}lm\vswfouttlm{\tau}lm\left(k\left(\vect r-\vect r_{p}\right)\right)\right),\label{eq:E field expansion multiparticle}\\
|
||||
& \vect r\in\openball{\vect r_{p}}{R_{p}^{>}}\backslash B_{\vect r_{p}}\left(R_{p}\right).\nonumber
|
||||
\end{align}
|
||||
|
||||
\end_inset
|
||||
|
||||
Unlike the single scatterer case, the incident field coefficients
|
||||
\begin_inset Formula $\rcoeffptlm p{\tau}lm$
|
||||
\end_inset
|
||||
|
||||
here are not only due to some external driving field that the particle
|
||||
does not influence but they also contain the contributions of fields scattered
|
||||
from
|
||||
\emph on
|
||||
all other scatterers
|
||||
\emph default
|
||||
:
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\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}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
where
|
||||
\begin_inset Formula $\rcoeffincp p$
|
||||
\end_inset
|
||||
|
||||
represents the part due to the external driving that the scatterers can
|
||||
not influence, and
|
||||
\begin_inset Formula $\tropsp pq$
|
||||
\end_inset
|
||||
|
||||
is a
|
||||
\emph on
|
||||
translation operator
|
||||
\emph default
|
||||
defined below in Sec.
|
||||
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
reference "subsec:Translation-operator"
|
||||
plural "false"
|
||||
caps "false"
|
||||
noprefix "false"
|
||||
|
||||
\end_inset
|
||||
|
||||
, that contains the re-expansion coefficients of the outgoing waves in origin
|
||||
|
||||
\begin_inset Formula $\vect r_{q}$
|
||||
\end_inset
|
||||
|
||||
into regular waves in origin
|
||||
\begin_inset Formula $\vect r_{p}$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Subsubsection
|
||||
Translation operator
|
||||
\begin_inset CommandInset label
|
||||
LatexCommand label
|
||||
name "subsec:Translation-operator"
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Let
|
||||
\begin_inset Formula $\vect r_{1},\vect r_{2}$
|
||||
\end_inset
|
||||
|
||||
be two different origins; a regular VSWF with origin
|
||||
\begin_inset Formula $\vect r_{1}$
|
||||
\end_inset
|
||||
|
||||
can be always expanded in terms of regular VSWFs with origin
|
||||
\begin_inset Formula $\vect r_{2}$
|
||||
\end_inset
|
||||
|
||||
as follows:
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\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}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
where an explicit formula for the (regular)
|
||||
\emph on
|
||||
translation operator
|
||||
\emph default
|
||||
|
||||
\begin_inset Formula $\tropr$
|
||||
\end_inset
|
||||
|
||||
reads in eq.
|
||||
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:translation operator"
|
||||
|
||||
\end_inset
|
||||
|
||||
below.
|
||||
For singular (outgoing) waves, the form of the expansion differs inside
|
||||
and outside the ball
|
||||
\begin_inset Formula $\openball{\left\Vert \vect r_{2}-\vect r_{1}\right\Vert }{\vect r_{1}}:$
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Formula
|
||||
\begin{eqnarray}
|
||||
\vswfouttlm{\tau}lm\left(k\left(\vect r-\vect r_{1}\right)\right) & = & \begin{cases}
|
||||
\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}}\\
|
||||
\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|}
|
||||
\end{cases},\label{eq:singular vswf translation}
|
||||
\end{eqnarray}
|
||||
|
||||
\end_inset
|
||||
|
||||
where the singular translation operator
|
||||
\begin_inset Formula $\trops$
|
||||
\end_inset
|
||||
|
||||
has the same form as
|
||||
\begin_inset Formula $\tropr$
|
||||
\end_inset
|
||||
|
||||
in
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:translation operator"
|
||||
|
||||
\end_inset
|
||||
|
||||
except the regular spherical Bessel functions
|
||||
\begin_inset Formula $j_{l}$
|
||||
\end_inset
|
||||
|
||||
are replaced with spherical Hankel functions
|
||||
\begin_inset Formula $h_{l}^{(1)}$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
TODO note about expansion exactly on the sphere.
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
As MSTMM deals most of the time with the
|
||||
\emph on
|
||||
expansion coefficients
|
||||
\emph default
|
||||
of fields
|
||||
\begin_inset Formula $\rcoeffptlm p{\tau}lm,\outcoeffptlm p{\tau}lm$
|
||||
\end_inset
|
||||
|
||||
in different origins
|
||||
\begin_inset Formula $\vect r_{p}$
|
||||
\end_inset
|
||||
|
||||
rather than with the VSWFs directly, let us write down how
|
||||
\emph on
|
||||
they
|
||||
\emph default
|
||||
transform under translation.
|
||||
Let us assume the field can be in terms of regular waves everywhere, and
|
||||
expand it in two different origins
|
||||
\begin_inset Formula $\vect r_{p},\vect r_{q}$
|
||||
\end_inset
|
||||
|
||||
,
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\vect E\left(\vect r,\omega\right)=\sum_{\tau,l,m}\rcoeffptlm p{\tau}lm\vswfrtlm{\tau}lm\left(k\left(\vect r-\vect r_{p}\right)\right)=\sum_{\tau',l',m'}\rcoeffptlm q{\tau'}{l'}{m'}\vswfrtlm{\tau}{'l'}{m'}\left(k\left(\vect r-\vect r_{q}\right)\right).
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
Re-expanding the waves around
|
||||
\begin_inset Formula $\vect r_{p}$
|
||||
\end_inset
|
||||
|
||||
in terms of waves around
|
||||
\begin_inset Formula $\vect r_{q}$
|
||||
\end_inset
|
||||
|
||||
using
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:regular vswf translation"
|
||||
|
||||
\end_inset
|
||||
|
||||
,
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\vect E\left(\vect r,\omega\right)=\sum_{\tau,l,m}\rcoeffptlm p{\tau}lm\sum_{\tau'l'm'}\tropr_{\tau lm;\tau'l'm'}\left(k\left(\vect r_{q}-\vect r_{p}\right)\right)\vswfrtlm{\tau'}{l'}{m'}\left(\vect r-\vect r_{q}\right)
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
and comparing to the original expansion around
|
||||
\begin_inset Formula $\vect r_{q}$
|
||||
\end_inset
|
||||
|
||||
, we obtain
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\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}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
For the sake of readability, we introduce a shorthand matrix form for
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:regular vswf coefficient translation"
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\rcoeffp q=\troprp qp\rcoeffp p\label{eq:reqular vswf coefficient vector translation}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
(note the reversed indices; TODO redefine them in
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:regular vswf translation"
|
||||
|
||||
\end_inset
|
||||
|
||||
,
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:singular vswf translation"
|
||||
|
||||
\end_inset
|
||||
|
||||
? Similarly, if we had only outgoing waves in the original expansion around
|
||||
|
||||
\begin_inset Formula $\vect r_{p}$
|
||||
\end_inset
|
||||
|
||||
, we would get
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\rcoeffp q=\tropsp qp\outcoeffp p\label{eq:singular to regular vswf coefficient vector translation}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
for the expansion inside the ball
|
||||
\begin_inset Formula $\openball{\left\Vert \vect r_{q}-\vect r_{p}\right\Vert }{\vect r_{p}}$
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
CHECKME
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
and
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\outcoeffp q=\troprp qp\outcoeffp p\label{eq:singular to singular vswf coefficient vector translation-1}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
outside.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
In our convention, the regular translation operator can be expressed explicitly
|
||||
as
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right)=\dots.\label{eq:translation operator}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
The singular operator
|
||||
\begin_inset Formula $\trops$
|
||||
\end_inset
|
||||
|
||||
for re-expanding outgoing waves into regular ones has the same form except
|
||||
the regular spherical Bessel functions
|
||||
\begin_inset Formula $j_{l}$
|
||||
\end_inset
|
||||
|
||||
in are replaced with spherical Hankel functions
|
||||
\begin_inset Formula $h_{l}^{(1)}=j_{l}+iy_{l}$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
In our convention, the regular translation operator is unitary,
|
||||
\begin_inset Formula $\left(\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right)\right)^{-1}=\tropr_{\tau lm;\tau'l'm'}\left(-\vect d\right)=\tropr_{\tau'l'm';\tau lm}^{*}\left(\vect d\right)$
|
||||
\end_inset
|
||||
|
||||
,
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
todo different notation for the complex conjugation without transposition???
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
or in the per-particle matrix notation,
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\troprp qp^{-1}=\troprp pq=\troprp qp^{\dagger}\label{eq:regular translation unitarity}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
.
|
||||
Note that truncation at finite multipole degree breaks the unitarity,
|
||||
\begin_inset Formula $\truncated{\troprp qp}L^{-1}\ne\truncated{\troprp pq}L=\truncated{\troprp qp^{\dagger}}L$
|
||||
\end_inset
|
||||
|
||||
, which has to be taken into consideration when evaluating quantities such
|
||||
as absorption or scattering cross sections.
|
||||
Similarly, the full regular operators can be composed
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
better wording
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
,
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\troprp ac=\troprp ab\troprp bc\label{eq:regular translation composition}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
but truncation breaks this,
|
||||
\begin_inset Formula $\truncated{\troprp ac}L\ne\truncated{\troprp ab}L\truncated{\troprp bc}L.$
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Subsubsection
|
||||
Cross-sections (many scatterers)
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
For a system of many scatterers, Kristensson
|
||||
\begin_inset CommandInset citation
|
||||
LatexCommand cite
|
||||
after "sect. 9.2.2"
|
||||
key "kristensson_scattering_2016"
|
||||
literal "false"
|
||||
|
||||
\end_inset
|
||||
|
||||
derives only the extinction cross section formula.
|
||||
Let us re-derive it together with the many-particle scattering and absorption
|
||||
cross sections.
|
||||
First, let us take a ball circumscribing all the scatterers at once,
|
||||
\begin_inset Formula $\openball R{\vect r_{\square}}\supset\particle$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
Outside
|
||||
\begin_inset Formula $\openball R{\vect r_{\square}}$
|
||||
\end_inset
|
||||
|
||||
, we can describe the EM fields as if there was only a single scatterer,
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\vect E\left(\vect r\right)=\sum_{\tau,l,m}\left(\rcoeffptlm{\square}{\tau}lm\vswfrtlm{\tau}lm\left(\vect r-\vect r_{\square}\right)+\outcoeffptlm{\square}{\tau}lm\vswfouttlm{\tau}lm\left(\vect r-\vect r_{\square}\right)\right),
|
||||
\]
|
||||
|
||||
\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
|
||||
|
||||
with
|
||||
\begin_inset Formula $\rcoeffp{}=\rcoeffp{\square},\outcoeffp{}=\outcoeffp{\square}$
|
||||
\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
|
||||
|
||||
.
|
||||
The original incident field re-expanded around
|
||||
\begin_inset Formula $p$
|
||||
\end_inset
|
||||
|
||||
-th particle reads according to
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
reference "eq:regular vswf translation"
|
||||
plural "false"
|
||||
caps "false"
|
||||
noprefix "false"
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\rcoeffincp p=\troprp p{\square}\rcoeffp{\square}\label{eq:a_inc local from global}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
whereas the contributions of fields scattered from each particle expanded
|
||||
around the global origin
|
||||
\begin_inset Formula $\vect r_{\square}$
|
||||
\end_inset
|
||||
|
||||
is, according to
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
reference "eq:singular vswf translation"
|
||||
plural "false"
|
||||
caps "false"
|
||||
noprefix "false"
|
||||
|
||||
\end_inset
|
||||
|
||||
,
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\outcoeffp{\square}=\sum_{q\in\mathcal{P}}\troprp{\square}q\outcoeffp q.\label{eq:f global from local}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
Using the unitarity
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
reference "eq:regular translation unitarity"
|
||||
plural "false"
|
||||
caps "false"
|
||||
noprefix "false"
|
||||
|
||||
\end_inset
|
||||
|
||||
and composition
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
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 \\
|
||||
\label{eq:absorption CS multi}
|
||||
\end{eqnarray}
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
\begin_inset Formula $=\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\sum_{p\in\mathcal{P}}\mbox{TODO}.$
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\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
|
||||
|
||||
\begin_layout Subsubsection
|
||||
|
|
|
|||
Loading…
Reference in New Issue