Single particle scattering progress.
Former-commit-id: f0a3cf7f42f95b36a9f4db4118c06dc1b3d737c6
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Marek Nečada
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9aefacfa00
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@ -279,6 +279,11 @@
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\rcoefftlm}[3]{\rcoeffp{#1#2#3}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\rcoeffincptlm}[4]{\rcoeffincp{#1,#2#3#4}}
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\end_inset
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@ -299,6 +304,11 @@
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\outcoefftlm}[3]{\outcoeffp{#1#2#3}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\vswfouttlm}[3]{\vect u_{#1#2#3}}
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\end_inset
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@ -344,6 +354,11 @@
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\dlmfFer}[2]{\mathsf{P}_{#1}^{#2}}
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\end_inset
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\end_layout
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\begin_layout Standard
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@ -405,10 +405,19 @@ literal "false"
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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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Its solutions (TODO under which conditions? What vector space do the SVWFs
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actually span? Check Comment 9.2 and Appendix f.9.1 in Kristensson)
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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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\begin_inset Note Note
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status open
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@ -427,14 +436,284 @@ TODO small note about cartesian multipoles, anapoles etc.
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T-matrix definition
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\end_layout
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\begin_layout Subsubsection
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Absorbed power
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\begin_layout Standard
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The regular VSWFs
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\begin_inset Formula $\vswfrtlm{\tau}lm\left(k\vect r\right)$
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\end_inset
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constitute a basis for solutions of the Helmholtz equation
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:Helmholtz eq"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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inside a ball
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\begin_inset Formula $\openball 0R$
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\end_inset
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with radius
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\begin_inset Formula $R$
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\end_inset
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and center in the origin; however, if the equation is not guaranteed to
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hold inside a smaller ball
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\begin_inset Formula $B_{0}\left(R_{<}\right)$
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\end_inset
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around the origin (typically due to presence of a scatterer), one has to
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add the outgoing VSWFs
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\begin_inset Formula $\vswfrtlm{\tau}lm\left(k\vect r\right)$
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\end_inset
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to have a complete basis of the solutions in the volume
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\begin_inset Formula $\openball 0R\backslash B_{0}\left(R_{<}\right)$
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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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Vnitřní koule uzavřená? Jak se řekne mezikulí anglicky?
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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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The single-particle scattering problem at frequency
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\begin_inset Formula $\omega$
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\end_inset
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can be posed as follows: Let a scatterer be enclosed inside the ball
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\begin_inset Formula $B_{0}\left(R_{<}\right)$
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\end_inset
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and let the whole volume
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\begin_inset Formula $\openball 0R\backslash B_{0}\left(R_{<}\right)$
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\end_inset
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be filled with a homogeneous isotropic medium with wave number
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\begin_inset Formula $k\left(\omega\right)$
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\end_inset
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.
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Inside this volume, the electric field can be expanded as
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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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doplnit frekvence a polohy
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\end_layout
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\end_inset
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\begin_inset Formula
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\[
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\vect E\left(\omega,\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\left(\rcoefftlm{\tau}lm\vswfrtlm{\tau}lm+\outcoefftlm{\tau}lm\vswfouttlm{\tau}lm\right).
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\]
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\end_inset
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If there was no scatterer and
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\begin_inset Formula $B_{0}\left(R_{<}\right)$
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\end_inset
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was filled with the same homogeneous medium, the part with the outgoing
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VSWFs would vanish and only the part
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\begin_inset Formula $\vect E_{\mathrm{inc}}=\sum_{\tau lm}\rcoefftlm{\tau}lm\vswfrtlm{\tau}lm$
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\end_inset
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due to sources outside
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\begin_inset Formula $\openball 0R$
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\end_inset
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would remain.
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Let us assume that the
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\begin_inset Quotes eld
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\end_inset
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driving field
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\begin_inset Quotes erd
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\end_inset
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is given, so that presence of the scatterer does not affect
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\begin_inset Formula $\vect E_{\mathrm{inc}}$
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\end_inset
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and is fully manifested in the latter part,
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\begin_inset Formula $\vect E_{\mathrm{scat}}=\sum_{\tau lm}\outcoefftlm{\tau}lm\vswfouttlm{\tau}lm$
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\end_inset
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.
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We also assume that the scatterer is made of optically linear materials,
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and hence reacts on the incident field in a linear manner.
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This gives a linearity constraint between the expansion coefficients
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\begin_inset Formula
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\begin{equation}
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\outcoefftlm{\tau}lm=\sum_{\tau'l'm'}T_{\tau lm}^{\tau'l'm'}\rcoefftlm{\tau'}{l'}{m'}\label{eq:T-matrix definition}
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\end{equation}
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\end_inset
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where the
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\begin_inset Formula $T_{\tau lm}^{\tau'l'm'}=T_{\tau lm}^{\tau'l'm'}\left(\omega\right)$
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\end_inset
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are the elements of the
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\emph on
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transition matrix,
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\emph default
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a.k.a.
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\begin_inset Formula $T$
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\end_inset
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-matrix.
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It completely describes the scattering properties of a linear scatterer,
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so with the knowledge of the
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\begin_inset Formula $T$
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\end_inset
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-matrix, we can solve the single-patricle scatering prroblem simply by substitut
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ing appropriate expansion coefficients
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\begin_inset Formula $\rcoefftlm{\tau'}{l'}{m'}$
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\end_inset
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of the driving field into
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:T-matrix definition"
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plural "false"
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caps "false"
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noprefix "false"
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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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\begin_inset Formula $T$
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\end_inset
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-matrices of particles with certain simple geometries (most famously spherical)
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can be obtained analytically [Kristensson 2016, Mie], but in general one
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can find them numerically by simulating scattering of a regular spherical
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wave
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\begin_inset Formula $\vswfouttlm{\tau}lm$
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\end_inset
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and projecting the scattered fields (or induced currents, depending on
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the method) onto the outgoing VSWFs
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\begin_inset Formula $\vswfrtlm{\tau}{'l'}{m'}$
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\end_inset
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.
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In practice, one can compute only a finite number of elements with a cut-off
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value
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\begin_inset Formula $L$
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\end_inset
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on the multipole degree,
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\begin_inset Formula $l,l'\le L$
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\end_inset
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, see below.
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We typically use the scuff-tmatrix tool from the free software SCUFF-EM
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suite [SCUFF-EM].
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Note that older versions of SCUFF-EM contained a bug that rendered almost
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all
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\begin_inset Formula $T$
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\end_inset
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-matrix results wrong; we found and fixed the bug and from upstream version
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xxx onwards, it should behave correctly.
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\end_layout
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\begin_layout Subsubsection
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T-matrix compactness, cutoff validity
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\end_layout
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\begin_layout Standard
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The magnitude of the
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\begin_inset Formula $T$
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\end_inset
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-matrix elements depends heavily on the scatterer's size compared to the
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wavelength.
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Fortunately, the
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\begin_inset Formula $T$
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\end_inset
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-matrix of a bounded scatterer is a compact operator [REF???], so from certain
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multipole degree onwards,
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\begin_inset Formula $l,l'>L$
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\end_inset
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, the elements of the
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\begin_inset Formula $T$
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\end_inset
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-matrix are negligible, so truncating the
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\begin_inset Formula $T$
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\end_inset
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-matrix at finite multipole degree
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\begin_inset Formula $L$
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\end_inset
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gives a good approximation of the actual infinite-dimensional itself.
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If the incident field is well-behaved, i.e.
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the expansion coefficients
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\begin_inset Formula $\rcoefftlm{\tau'}{l'}{m'}$
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\end_inset
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do not take excessive values for
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\begin_inset Formula $l'>L$
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\end_inset
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, the scattered field expansion coefficients
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\begin_inset Formula $\outcoefftlm{\tau}lm$
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\end_inset
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with
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\begin_inset Formula $l>L$
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\end_inset
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will also be negligible.
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\end_layout
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\begin_layout Standard
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A rule of thumb to choose the
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\begin_inset Formula $L$
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\end_inset
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with desired
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\begin_inset Formula $T$
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\end_inset
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-matrix element accuracy
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\begin_inset Formula $\delta$
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\end_inset
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can be obtained from the spherical Bessel function expansion around zero,
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TODO.
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\end_layout
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\begin_layout Subsubsection
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Absorbed power
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\end_layout
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\begin_layout Subsection
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Multiple scattering
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\end_layout
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