From 80ea82a33f46f8a71179ac638401be20b3eb7674 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Mon, 29 Jul 2019 10:14:08 +0300 Subject: [PATCH] Single particle scattering progress. Former-commit-id: f0a3cf7f42f95b36a9f4db4118c06dc1b3d737c6 --- lepaper/arrayscat.lyx | 15 +++ lepaper/finite.lyx | 283 +++++++++++++++++++++++++++++++++++++++++- 2 files changed, 296 insertions(+), 2 deletions(-) diff --git a/lepaper/arrayscat.lyx b/lepaper/arrayscat.lyx index 01b2c2e..01e1e6c 100644 --- a/lepaper/arrayscat.lyx +++ b/lepaper/arrayscat.lyx @@ -279,6 +279,11 @@ \end_inset +\begin_inset FormulaMacro +\newcommand{\rcoefftlm}[3]{\rcoeffp{#1#2#3}} +\end_inset + + \begin_inset FormulaMacro \newcommand{\rcoeffincptlm}[4]{\rcoeffincp{#1,#2#3#4}} \end_inset @@ -299,6 +304,11 @@ \end_inset +\begin_inset FormulaMacro +\newcommand{\outcoefftlm}[3]{\outcoeffp{#1#2#3}} +\end_inset + + \begin_inset FormulaMacro \newcommand{\vswfouttlm}[3]{\vect u_{#1#2#3}} \end_inset @@ -344,6 +354,11 @@ \end_inset +\begin_inset FormulaMacro +\newcommand{\dlmfFer}[2]{\mathsf{P}_{#1}^{#2}} +\end_inset + + \end_layout \begin_layout Standard diff --git a/lepaper/finite.lyx b/lepaper/finite.lyx index 7f06958..1f8c54c 100644 --- a/lepaper/finite.lyx +++ b/lepaper/finite.lyx @@ -405,10 +405,19 @@ literal "false" \end_layout \begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout Its solutions (TODO under which conditions? What vector space do the SVWFs actually span? Check Comment 9.2 and Appendix f.9.1 in Kristensson) \end_layout +\end_inset + + +\end_layout + \begin_layout Standard \begin_inset Note Note status open @@ -427,14 +436,284 @@ TODO small note about cartesian multipoles, anapoles etc. T-matrix definition \end_layout -\begin_layout Subsubsection -Absorbed power +\begin_layout Standard +The regular VSWFs +\begin_inset Formula $\vswfrtlm{\tau}lm\left(k\vect r\right)$ +\end_inset + + constitute a basis for solutions of the Helmholtz equation +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:Helmholtz eq" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + inside a ball +\begin_inset Formula $\openball 0R$ +\end_inset + + with radius +\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 $B_{0}\left(R_{<}\right)$ +\end_inset + + around the origin (typically due to presence of a scatterer), one has to + add the outgoing VSWFs +\begin_inset Formula $\vswfrtlm{\tau}lm\left(k\vect r\right)$ +\end_inset + + to have a complete basis of the solutions in the volume +\begin_inset Formula $\openball 0R\backslash B_{0}\left(R_{<}\right)$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +Vnitřní koule uzavřená? Jak se řekne mezikulí anglicky? +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +The single-particle scattering problem at frequency +\begin_inset Formula $\omega$ +\end_inset + + can be posed as follows: Let a scatterer be enclosed inside the ball +\begin_inset Formula $B_{0}\left(R_{<}\right)$ +\end_inset + + and let the whole volume +\begin_inset Formula $\openball 0R\backslash B_{0}\left(R_{<}\right)$ +\end_inset + + be filled with a homogeneous isotropic medium with wave number +\begin_inset Formula $k\left(\omega\right)$ +\end_inset + +. + Inside this volume, the electric field can be expanded as +\begin_inset Note Note +status open + +\begin_layout Plain Layout +doplnit frekvence a polohy +\end_layout + +\end_inset + + +\begin_inset Formula +\[ +\vect E\left(\omega,\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\left(\rcoefftlm{\tau}lm\vswfrtlm{\tau}lm+\outcoefftlm{\tau}lm\vswfouttlm{\tau}lm\right). +\] + +\end_inset + +If there was no scatterer and +\begin_inset Formula $B_{0}\left(R_{<}\right)$ +\end_inset + + was filled with the same homogeneous medium, the part with the outgoing + VSWFs would vanish and only the part +\begin_inset Formula $\vect E_{\mathrm{inc}}=\sum_{\tau lm}\rcoefftlm{\tau}lm\vswfrtlm{\tau}lm$ +\end_inset + + due to sources outside +\begin_inset Formula $\openball 0R$ +\end_inset + + would remain. + Let us assume that the +\begin_inset Quotes eld +\end_inset + +driving field +\begin_inset Quotes erd +\end_inset + + is given, so that presence of the scatterer does not affect +\begin_inset Formula $\vect E_{\mathrm{inc}}$ +\end_inset + + and is fully manifested in the latter part, +\begin_inset Formula $\vect E_{\mathrm{scat}}=\sum_{\tau lm}\outcoefftlm{\tau}lm\vswfouttlm{\tau}lm$ +\end_inset + +. + We also assume that the scatterer is made of optically linear materials, + and hence reacts on the incident field in a linear manner. + This gives a linearity constraint between the expansion coefficients +\begin_inset Formula +\begin{equation} +\outcoefftlm{\tau}lm=\sum_{\tau'l'm'}T_{\tau lm}^{\tau'l'm'}\rcoefftlm{\tau'}{l'}{m'}\label{eq:T-matrix definition} +\end{equation} + +\end_inset + +where the +\begin_inset Formula $T_{\tau lm}^{\tau'l'm'}=T_{\tau lm}^{\tau'l'm'}\left(\omega\right)$ +\end_inset + + are the elements of the +\emph on +transition matrix, +\emph default + a.k.a. + +\begin_inset Formula $T$ +\end_inset + +-matrix. + It completely describes the scattering properties of a linear scatterer, + so with the knowledge of the +\begin_inset Formula $T$ +\end_inset + +-matrix, we can solve the single-patricle scatering prroblem simply by substitut +ing appropriate expansion coefficients +\begin_inset Formula $\rcoefftlm{\tau'}{l'}{m'}$ +\end_inset + + of the driving field into +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:T-matrix definition" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $T$ +\end_inset + +-matrices of particles with certain simple geometries (most famously spherical) + can be obtained analytically [Kristensson 2016, Mie], but in general one + can find them numerically by simulating scattering of a regular spherical + wave +\begin_inset Formula $\vswfouttlm{\tau}lm$ +\end_inset + + and projecting the scattered fields (or induced currents, depending on + the method) onto the outgoing VSWFs +\begin_inset Formula $\vswfrtlm{\tau}{'l'}{m'}$ +\end_inset + +. + In practice, one can compute only a finite number of elements with a cut-off + value +\begin_inset Formula $L$ +\end_inset + + on the multipole degree, +\begin_inset Formula $l,l'\le L$ +\end_inset + +, see below. + We typically use the scuff-tmatrix tool from the free software SCUFF-EM + suite [SCUFF-EM]. + Note that older versions of SCUFF-EM contained a bug that rendered almost + all +\begin_inset Formula $T$ +\end_inset + +-matrix results wrong; we found and fixed the bug and from upstream version + xxx onwards, it should behave correctly. + \end_layout \begin_layout Subsubsection T-matrix compactness, cutoff validity \end_layout +\begin_layout Standard +The magnitude of the +\begin_inset Formula $T$ +\end_inset + +-matrix elements depends heavily on the scatterer's size compared to the + wavelength. + Fortunately, the +\begin_inset Formula $T$ +\end_inset + +-matrix of a bounded scatterer is a compact operator [REF???], so from certain + multipole degree onwards, +\begin_inset Formula $l,l'>L$ +\end_inset + +, the elements of the +\begin_inset Formula $T$ +\end_inset + +-matrix are negligible, so truncating the +\begin_inset Formula $T$ +\end_inset + +-matrix at finite multipole degree +\begin_inset Formula $L$ +\end_inset + + gives a good approximation of the actual infinite-dimensional itself. + If the incident field is well-behaved, i.e. + the expansion coefficients +\begin_inset Formula $\rcoefftlm{\tau'}{l'}{m'}$ +\end_inset + + do not take excessive values for +\begin_inset Formula $l'>L$ +\end_inset + +, the scattered field expansion coefficients +\begin_inset Formula $\outcoefftlm{\tau}lm$ +\end_inset + + with +\begin_inset Formula $l>L$ +\end_inset + + will also be negligible. +\end_layout + +\begin_layout Standard +A rule of thumb to choose the +\begin_inset Formula $L$ +\end_inset + + with desired +\begin_inset Formula $T$ +\end_inset + +-matrix element accuracy +\begin_inset Formula $\delta$ +\end_inset + + can be obtained from the spherical Bessel function expansion around zero, + TODO. + +\end_layout + +\begin_layout Subsubsection +Absorbed power +\end_layout + \begin_layout Subsection Multiple scattering \end_layout