From e26ca79a53c7e0e39a6250b3a3a67f1a22f1c01b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Sat, 20 Jul 2019 13:54:17 +0300 Subject: [PATCH] WSWF translations (TODO explicit form of the ops) Former-commit-id: 44f286281af3f44208c2b1e2aabbe7234b0cd675 --- lepaper/arrayscat.lyx | 92 +++++++- lepaper/finite-cs.lyx | 504 ++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 595 insertions(+), 1 deletion(-) create mode 100644 lepaper/finite-cs.lyx diff --git a/lepaper/arrayscat.lyx b/lepaper/arrayscat.lyx index ab54aa2..7bbf6c2 100644 --- a/lepaper/arrayscat.lyx +++ b/lepaper/arrayscat.lyx @@ -117,6 +117,11 @@ \end_inset +\begin_inset FormulaMacro +\newcommand{\uvec}[1]{\mathbf{\hat{#1}}} +\end_inset + + \begin_inset FormulaMacro \newcommand{\ud}{\mathrm{d}} \end_inset @@ -162,6 +167,16 @@ \end_inset +\begin_inset FormulaMacro +\newcommand{\vsh}[3]{\vect A_{#1,#2,#3}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\vshD}[3]{\vect A'_{#1,#2,#3}} +\end_inset + + \begin_inset FormulaMacro \newcommand{\hgfr}{\mathbf{F}} \end_inset @@ -213,7 +228,7 @@ \begin_inset FormulaMacro -\newcommand{\particle}{\mathrm{\Omega}} +\newcommand{\particle}{\mathrm{\Theta}} \end_inset @@ -232,6 +247,71 @@ \end_inset +\begin_inset FormulaMacro +\newcommand{\rcoeffp}[1]{a_{#1}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\rcoeffptlm}[4]{\rcoeffp{#1}_{#2#3#4}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\vswfrtlm}[3]{\vect v_{#1#2#3}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\outcoeffp}[1]{f_{#1}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\outcoeffptlm}[4]{\outcoeffp{#1}_{#2#3#4}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\vswfouttlm}[3]{\vect u_{#1#2#3}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\Tp}[1]{T_{#1}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\openball}[2]{B_{#1}\left(#2\right)} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\closedball}[2]{B_{#1}#2} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\tropr}{\mathcal{R}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\troprp}[2]{\mathcal{\tropr}_{#1\leftarrow#2}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\trops}{\mathcal{S}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\tropsp}[2]{\mathcal{\trops}_{#1\leftarrow#2}} +\end_inset + + \end_layout \begin_layout Title @@ -450,6 +530,16 @@ filename "finite.lyx" \end_inset +\end_layout + +\begin_layout Standard +\begin_inset CommandInset include +LatexCommand include +filename "finite-cs.lyx" + +\end_inset + + \end_layout \begin_layout Standard diff --git a/lepaper/finite-cs.lyx b/lepaper/finite-cs.lyx new file mode 100644 index 0000000..5387b2b --- /dev/null +++ b/lepaper/finite-cs.lyx @@ -0,0 +1,504 @@ +#LyX 2.1 created this file. For more info see http://www.lyx.org/ +\lyxformat 474 +\begin_document +\begin_header +\textclass article +\use_default_options true +\maintain_unincluded_children false +\language finnish +\language_package default +\inputencoding auto +\fontencoding global +\font_roman TeX Gyre Pagella +\font_sans default +\font_typewriter default +\font_math auto +\font_default_family default +\use_non_tex_fonts true +\font_sc false +\font_osf true +\font_sf_scale 100 +\font_tt_scale 100 +\graphics default +\default_output_format pdf4 +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize default +\spacing single +\use_hyperref true +\pdf_title "Sähköpajan päiväkirja" +\pdf_author "Marek Nečada" +\pdf_bookmarks true +\pdf_bookmarksnumbered false +\pdf_bookmarksopen false +\pdf_bookmarksopenlevel 1 +\pdf_breaklinks false +\pdf_pdfborder false +\pdf_colorlinks false +\pdf_backref false +\pdf_pdfusetitle true +\papersize default +\use_geometry false +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification true +\use_refstyle 1 +\index Index +\shortcut idx +\color #008000 +\end_index +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\quotes_language swedish +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Subsection +Dual vector spherical harmonics +\end_layout + +\begin_layout Standard +For evaluation of expansion coefficients of incident fields, it is useful + to introduce „dual“ vector spherical harmonics +\begin_inset Formula $\vshD{\tau}lm$ +\end_inset + + defined by duality relation +\begin_inset Formula +\begin{equation} +\iint\vshD{\tau'}{l'}{m'}\left(\uvec r\right)\cdot\vsh{\tau}lm\left(\uvec r\right)\,\ud\Omega=\delta_{\tau'\tau}\delta_{l'l}\delta_{m'm}\label{eq:dual vsh} +\end{equation} + +\end_inset + +(complex conjugation not implied in the dot product here). + In our convention, we have +\begin_inset Formula +\[ +\vshD{\tau}lm\left(\uvec r\right)=\left(\vsh{\tau}lm\left(\uvec r\right)\right)^{*}=\left(-1\right)^{m}\vsh{\tau}{l-}m\left(\uvec r\right). +\] + +\end_inset + + +\end_layout + +\begin_layout Subsection +Translation operators +\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 +The 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 + + +\end_layout + +\begin_layout Subsection +Plane wave expansion coefficients +\end_layout + +\begin_layout Standard +A transversal ( +\begin_inset Formula $\vect k\cdot\vect E_{0}=0$ +\end_inset + +) plane wave propagating in direction +\begin_inset Formula $\uvec k$ +\end_inset + + with (complex) amplitude +\begin_inset Formula $\vect E_{0}$ +\end_inset + + can be expanded into regular VSWFs [REF KRIS] +\begin_inset Formula +\[ +\vect E_{\mathrm{PW}}\left(\vect r,\omega\right)=\vect E_{0}e^{ik\uvec k\cdot\vect r}=\sum_{\tau,l,m}\rcoeffptlm{}{\tau}lm\left(\vect k,\vect E_{0}\right)\vswfrtlm{\tau}lm\left(k\vect r\right), +\] + +\end_inset + +with expansion coefficients +\begin_inset Formula +\begin{eqnarray} +\rcoeffptlm{}1lm\left(\vect k,\vect E_{0}\right) & = & 4\pi i^{l}\vshD 1lm\left(\uvec k\right),\nonumber \\ +\rcoeffptlm{}2lm\left(\vect k,\vect E_{0}\right) & = & -4\pi i^{l+1}\vshD 2lm\left(\uvec k\right).\label{eq:plane wave expansion} +\end{eqnarray} + +\end_inset + + +\end_layout + +\begin_layout Subsection +Cross-sections (single-particle) +\end_layout + +\begin_layout Standard +Extinction, scattering and absorption cross sections of a single particle + irradiated by a plane wave propagating in direction +\begin_inset Formula $\uvec k$ +\end_inset + + are +\begin_inset CommandInset citation +LatexCommand cite +after "sect. 7.8.2" +key "kristensson_scattering_2016" + +\end_inset + + +\begin_inset Formula +\begin{eqnarray} +\sigma_{\mathrm{ext}}\left(\uvec k\right) & = & -\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\Re\left(\rcoeffp{}^{\dagger}\outcoeffp{}\right)=-\frac{1}{2k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\rcoeffp{}^{\dagger}\left(\Tp{}+\Tp{}^{\dagger}\right)\rcoeffp{},\label{eq:extincion CS single}\\ +\sigma_{\mathrm{scat}}\left(\uvec k\right) & = & \frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\left\Vert \outcoeffp{}\right\Vert ^{2}=\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\rcoeffp{}^{\dagger}\left(\Tp{}^{\dagger}\Tp{}\right)\rcoeffp{},\label{eq:scattering CS single}\\ +\sigma_{\mathrm{abs}}\left(\uvec k\right) & = & \sigma_{\mathrm{ext}}\left(\uvec k\right)-\sigma_{\mathrm{scat}}\left(\uvec k\right)=-\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\left(\Re\left(\rcoeffp{}^{\dagger}\outcoeffp{}\right)+\left\Vert \outcoeffp{}\right\Vert ^{2}\right)\nonumber \\ + & & =\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\rcoeffp{}^{\dagger}\left(\Tp{}^{\dagger}\Tp{}+\frac{\Tp{}^{\dagger}+\Tp{}}{2}\right)\rcoeffp{},\label{eq:absorption CS single} +\end{eqnarray} + +\end_inset + +where +\begin_inset Formula $\rcoeffp{}=\rcoeffp{}\left(\vect k,\vect E_{0}\right)$ +\end_inset + + is the vector of plane wave expansion coefficients as in +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:plane wave expansion" + +\end_inset + +. +\end_layout + +\begin_layout Standard +For a system of many scatterers, Kristensson derives only the scattering + cross section formula +\begin_inset Formula +\[ +\sigma_{\mathrm{scat}}\left(\uvec k\right)=\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\sum_{p\in\mathcal{P}}\left\Vert \outcoeffp p\right\Vert ^{2}. +\] + +\end_inset + +Let us derive 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 + + to obtain the cross sections. + However, this is not suitable for numerical evaluation with truncation + in multipole degree; hence we need to express them in terms of particle-wise + expansions +\begin_inset Formula $\rcoeffp p,\outcoeffp p$ +\end_inset + +. +\end_layout + +\end_body +\end_document