2007 lines
50 KiB
Plaintext
2007 lines
50 KiB
Plaintext
#LyX 2.4 created this file. For more info see https://www.lyx.org/
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\pdf_author "Marek Nečada"
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\shortcut idx
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\end_header
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\begin_body
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\begin_layout Section
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Finite systems
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\end_layout
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\begin_layout Subsection
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Motivation/intro
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\end_layout
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\begin_layout Standard
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The basic idea of MSTMM is quite simple: the driving electromagnetic field
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incident onto a scatterer is expanded into a vector spherical wavefunction
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(VSWF) basis in which the single scattering problem is solved, and the
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scattered field is then re-expanded into VSWFs centered at the other scatterers.
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Repeating the same procedure with all (pairs of) scatterers yields a set
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of linear equations, solution of which gives the coefficients of the scattered
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field in the VSWF bases.
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Once these coefficients have been found, one can evaluate various quantities
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related to the scattering (such as cross sections or the scattered fields)
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quite easily.
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\end_layout
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\begin_layout Standard
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The expressions appearing in the re-expansions are fairly complicated, and
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the implementation of MSTMM is extremely error-prone also due to the various
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conventions used in the literature.
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Therefore although we do not re-derive from scratch the expressions that
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can be found elsewhere in literature, we always state them explicitly in
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our convention.
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\end_layout
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\begin_layout Subsection
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Single-particle scattering
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\end_layout
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\begin_layout Standard
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In order to define the basic concepts, let us first consider the case of
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EM radiation scattered by a single particle.
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We assume that the scatterer lies inside a closed sphere
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\begin_inset Formula $\particle$
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\end_inset
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, the space outside this volume
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\begin_inset Formula $\medium$
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\end_inset
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is filled with an homogeneous isotropic medium with relative electric permittiv
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ity
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\begin_inset Formula $\epsilon(\vect r,\omega)=\epsbg(\omega)$
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\end_inset
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and magnetic permeability
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\begin_inset Formula $\mu(\vect r,\omega)=\mubg(\omega)$
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\end_inset
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, and that the whole system is linear, i.e.
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the material properties of neither the medium nor the scatterer depend
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on field intensities.
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Under these assumptions, the EM fields
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\begin_inset Formula $\vect{\Psi}=\vect E,\vect H$
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\end_inset
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in
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\begin_inset Formula $\medium$
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\end_inset
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must satisfy the homogeneous vector Helmholtz equation together with the
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transversality condition
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\begin_inset Formula
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\begin{equation}
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\left(\nabla^{2}+k^{2}\right)\Psi\left(\vect r,\vect{\omega}\right)=0,\quad\nabla\cdot\vect{\Psi}\left(\vect r,\vect{\omega}\right)=0\label{eq:Helmholtz eq}
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\end{equation}
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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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frequency-space Maxwell's equations
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\begin_inset Formula
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\begin{align*}
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\nabla\times\vect E\left(\vect r,\omega\right)-ik\eta_{0}\eta\vect H\left(\vect r,\omega\right) & =0,\\
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\eta_{0}\eta\nabla\times\vect H\left(\vect r,\omega\right)+ik\vect E\left(\vect r,\omega\right) & =0.
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\end{align*}
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\end_inset
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\end_layout
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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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todo define
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\begin_inset Formula $\Psi$
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\end_inset
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, mention transversality
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\end_layout
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\end_inset
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with
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\begin_inset Formula $k=k\left(\omega\right)=\omega\sqrt{\mubg(\omega)\epsbg(\omega)}/c_{0}$
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\end_inset
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, as can be derived from the Maxwell's equations
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\begin_inset CommandInset citation
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LatexCommand cite
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after "???"
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key "jackson_classical_1998"
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literal "false"
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\end_inset
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.
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\end_layout
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\begin_layout Subsubsection
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Spherical waves
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\end_layout
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\begin_layout Standard
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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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can be solved by separation of variables in spherical coordinates to give
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the solutions – the
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\emph on
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regular
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\emph default
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and
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\emph on
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outgoing
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\emph default
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vector spherical wavefunctions (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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and
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\begin_inset Formula $\vswfouttlm{\tau}lm\left(k\vect r\right)$
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\end_inset
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, respectively, defined as follows:
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\begin_inset Formula
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\begin{align}
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\vswfrtlm 1lm\left(k\vect r\right) & =j_{l}\left(kr\right)\vsh 1lm\left(\uvec r\right),\nonumber \\
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\vswfrtlm 2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krj_{l}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(kr\right)}{kr}\vsh 3lm\left(\uvec r\right),\label{eq:VSWF regular}
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\end{align}
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\end_inset
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\begin_inset Formula
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\begin{align}
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\vswfouttlm 1lm\left(k\vect r\right) & =h_{l}^{\left(1\right)}\left(kr\right)\vsh 1lm\left(\uvec r\right),\nonumber \\
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\vswfouttlm 2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krh_{l}^{\left(1\right)}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{h_{l}^{\left(1\right)}\left(kr\right)}{kr}\vsh 3lm\left(\uvec r\right),\label{eq:VSWF outgoing}\\
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& \tau=1,2;\quad l=1,2,3,\dots;\quad m=-l,-l+1,\dots,+l,\nonumber
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\end{align}
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\end_inset
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where
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\begin_inset Formula $\vect r=r\uvec r$
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\end_inset
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,
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\begin_inset Formula $j_{l}\left(x\right),h_{l}^{\left(1\right)}\left(x\right)=j_{l}\left(x\right)+iy_{l}\left(x\right)$
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\end_inset
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are the regular spherical Bessel function and spherical Hankel function
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of the first kind, respectively, as in [DLMF §10.47], and
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\begin_inset Formula $\vsh{\tau}lm$
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\end_inset
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are the
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\emph on
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vector spherical harmonics
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\emph default
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\begin_inset Formula
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\begin{align}
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\vsh 1lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\times\left(\vect r\ush lm\left(\uvec r\right)\right)=\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\ush lm\left(\uvec r\right)\times\vect r,\nonumber \\
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\vsh 2lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}r\nabla\ush lm\left(\uvec r\right),\nonumber \\
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\vsh 3lm\left(\uvec r\right) & =\uvec r\ush lm\left(\uvec r\right).\label{eq:vector spherical harmonics definition}
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\end{align}
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\end_inset
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In our convention, the (scalar) spherical harmonics
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\begin_inset Formula $\ush lm$
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\end_inset
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are identical to those in [DLMF 14.30.1], i.e.
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\begin_inset Formula
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\[
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\ush lm=\left(\frac{\left(l-m\right)!\left(2l+1\right)}{4\pi\left(l+m\right)!}\right)^{\frac{1}{2}}e^{im\phi}\dlmfFer lm\left(\cos\theta\right)
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\]
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\end_inset
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where importantly, the Ferrers functions
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\begin_inset Formula $\dlmfFer lm$
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\end_inset
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defined as in [DLMF §14.3(i)] do already contain the Condon-Shortley phase
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\begin_inset Formula $\left(-1\right)^{m}$
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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 názornější definice.
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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 convention for VSWFs used here is the same as in
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\begin_inset CommandInset citation
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LatexCommand cite
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key "kristensson_spherical_2014"
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literal "false"
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\end_inset
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; over other conventions used elsewhere in literature, it has several fundamenta
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l advantages – most importantly, the translation operators introduced later
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in eq.
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:translation op def"
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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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are unitary, and it gives the simplest possible expressions for power transport
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and cross sections without additional
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\begin_inset Formula $l,m$
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\end_inset
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-dependent factors (for that reason, we also call our VSWFs as
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\emph on
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power-normalised
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\emph default
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).
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Power-normalisation and unitary translation operators are possible to achieve
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also with real spherical harmonics – such a convention is used in
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\begin_inset CommandInset citation
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LatexCommand cite
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key "kristensson_scattering_2016"
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literal "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 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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\begin_layout Plain Layout
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TODO small note about cartesian multipoles, anapoles etc.
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(There should be some comparing paper that the Russians at META 2018 mentioned.)
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Subsubsection
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T-matrix definition
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\end_layout
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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 eqref
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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 $\openball 0{R^{>}}$
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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 $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
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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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||
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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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||
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||
\end_inset
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||
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||
\end_layout
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||
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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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||
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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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||
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||
and let the whole volume
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||
\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
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||
\end_inset
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||
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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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.
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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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||
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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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||
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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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\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\left(k\vect r\right)+\outcoefftlm{\tau}lm\vswfouttlm{\tau}lm\left(k\vect r\right)\right).\label{eq:E field expansion}
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\end{equation}
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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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||
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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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||
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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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||
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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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||
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driving field
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\begin_inset Quotes erd
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||
\end_inset
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||
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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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||
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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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||
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||
\end_inset
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||
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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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||
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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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||
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||
\begin_inset Formula $T$
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||
\end_inset
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||
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||
-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 eqref
|
||
reference "eq:T-matrix definition"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
The outgoing VSWF expansion coefficients
|
||
\begin_inset Formula $\outcoefftlm{\tau}lm$
|
||
\end_inset
|
||
|
||
are the effective induced electric (
|
||
\begin_inset Formula $\tau=2$
|
||
\end_inset
|
||
|
||
) and magnetic (
|
||
\begin_inset Formula $\tau=1$
|
||
\end_inset
|
||
|
||
) multipole polarisation amplitudes of the scatterer, and this is why we
|
||
sometimes refer to the corresponding VSWFs as the electric and magnetic
|
||
VSWFs.
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
TODO mention the pseudovector character of magnetic VSWFs.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
TOOD H-field expansion here?
|
||
\end_layout
|
||
|
||
\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
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "kristensson_scattering_2016,mie_beitrage_1908"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
, 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
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "reid_efficient_2015,SCUFF2"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
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.
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
TODO when it will not be negligible
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\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
|
||
Power transport
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For convenience, let us introduce a short-hand matrix notation for the expansion
|
||
coefficients and related quantities, so that we do not need to write the
|
||
indices explicitly; so for example, eq.
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:T-matrix definition"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
would be written as
|
||
\begin_inset Formula $\outcoeffp{}=T\rcoeffp{}$
|
||
\end_inset
|
||
|
||
, where
|
||
\begin_inset Formula $\rcoeffp{},\outcoeffp{}$
|
||
\end_inset
|
||
|
||
are column vectors with the expansion coefficients.
|
||
Transposed and complex-conjugated matrices are labeled with the
|
||
\begin_inset Formula $\dagger$
|
||
\end_inset
|
||
|
||
superscript.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
With this notation, we state an important result about power transport,
|
||
derivation of which can be found in
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "sect. 7.3"
|
||
key "kristensson_scattering_2016"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
Let the field in
|
||
\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
|
||
\end_inset
|
||
|
||
have expansion as in
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:E field expansion"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
Then the net power transported from
|
||
\begin_inset Formula $B_{0}\left(R\right)$
|
||
\end_inset
|
||
|
||
to
|
||
\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
|
||
\end_inset
|
||
|
||
via by electromagnetic radiation is
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
P=\frac{1}{2k^{2}\eta_{0}\eta}\left(\Re\left(\rcoeffp{}^{\dagger}\outcoeffp{}\right)+\left\Vert \outcoeffp{}\right\Vert ^{2}\right)=\frac{1}{2k^{2}\eta_{0}\eta}\rcoeffp{}^{\dagger}\left(\Tp{}^{\dagger}\Tp{}+\frac{\Tp{}^{\dagger}+\Tp{}}{2}\right)\rcoeffp{}.\label{eq:Power transport}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
In realistic scattering setups, power is transferred by radiation into
|
||
\begin_inset Formula $B_{0}\left(R\right)$
|
||
\end_inset
|
||
|
||
and absorbed by the enclosed scatterer, so
|
||
\begin_inset Formula $P$
|
||
\end_inset
|
||
|
||
is negative and its magnitude equals to power absorbed by the scatterer.
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Plane wave expansion
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In many scattering problems considered in practice, the driving field is
|
||
a plane wave.
|
||
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
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "7.???"
|
||
key "kristensson_scattering_2016"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
as
|
||
\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
|
||
|
||
where
|
||
\begin_inset Formula $\vshD{\tau}lm$
|
||
\end_inset
|
||
|
||
are the
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
dual
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
vector spherical harmonics defined by duality relation with the
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
usual
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
vector spherical harmonics
|
||
\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 Subsubsection
|
||
Cross-sections (single-particle)
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
With the
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
-matrix and expansion coefficients of plane waves in hand, we can state
|
||
the expressions for cross-sections of a single scatterer.
|
||
Assuming a non-lossy background medium, extinction, scattering and absorption
|
||
cross sections of a single scatterer irradiated by a plane wave propagating
|
||
in direction
|
||
\begin_inset Formula $\uvec k$
|
||
\end_inset
|
||
|
||
and (complex) amplitude
|
||
\begin_inset Formula $\vect E_{0}$
|
||
\end_inset
|
||
|
||
are
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "sect. 7.8.2"
|
||
key "kristensson_scattering_2016"
|
||
literal "true"
|
||
|
||
\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 Subsection
|
||
Multiple scattering
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "subsec:Multiple-scattering"
|
||
|
||
\end_inset
|
||
|
||
|
||
\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 eqref
|
||
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
|
||
|
||
.
|
||
For each scatterer, we also have its
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
-matrix relation as in
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:T-matrix definition"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula
|
||
\[
|
||
\outcoeffp q=T_{q}\rcoeffp q.
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
Together with
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:particle total incident field coefficient a"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
, this gives rise to a set of linear equations
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\outcoeffp p-T_{p}\sum_{q\in\mathcal{P}\backslash\left\{ p\right\} }\tropsp pq\outcoeffp q=T_{p}\rcoeffincp p,\quad p\in\mathcal{P}\label{eq:Multiple-scattering problem}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
which defines the multiple-scattering problem.
|
||
If all the
|
||
\begin_inset Formula $p,q$
|
||
\end_inset
|
||
|
||
-indexed vectors and matrices (note that without truncation, they are infinite-d
|
||
imensional) are arranged into blocks of even larger vectors and matrices,
|
||
this can be written in a short-hand form
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\left(I-T\trops\right)\outcoeff=T\rcoeffinc\label{eq:Multiple-scattering problem block form}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $I$
|
||
\end_inset
|
||
|
||
is the identity matrix,
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
is a block-diagonal matrix containing all the individual
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
-matrices, and
|
||
\begin_inset Formula $\trops$
|
||
\end_inset
|
||
|
||
contains the individual
|
||
\begin_inset Formula $\tropsp pq$
|
||
\end_inset
|
||
|
||
matrices as the off-diagonal blocks, whereas the diagonal blocks are set
|
||
to zeros.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In practice, the multiple-scattering problem is solved in its truncated
|
||
form, in which all the
|
||
\begin_inset Formula $l$
|
||
\end_inset
|
||
|
||
-indices related to a given scatterer
|
||
\begin_inset Formula $p$
|
||
\end_inset
|
||
|
||
are truncated as
|
||
\begin_inset Formula $l\le L_{p}$
|
||
\end_inset
|
||
|
||
, laeving only
|
||
\begin_inset Formula $N_{p}=2L_{p}\left(L_{p}+2\right)$
|
||
\end_inset
|
||
|
||
different
|
||
\begin_inset Formula $\tau lm$
|
||
\end_inset
|
||
|
||
-multiindices left.
|
||
The truncation degree can vary for different scatterers (e.g.
|
||
due to different physical sizes), so the truncated block
|
||
\begin_inset Formula $\tropsp pq$
|
||
\end_inset
|
||
|
||
has shape
|
||
\begin_inset Formula $N_{p}\times N_{q}$
|
||
\end_inset
|
||
|
||
, not necessarily square.
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Such truncation of the translation operator
|
||
\begin_inset Formula $\tropsp pq$
|
||
\end_inset
|
||
|
||
is justified by the fact on the left, TODO
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
If no other type of truncation is done, there remain
|
||
\begin_inset Formula $2L_{p}\left(L_{p}+2\right)$
|
||
\end_inset
|
||
|
||
different
|
||
\begin_inset Formula $\tau lm$
|
||
\end_inset
|
||
|
||
-multiindices for
|
||
\begin_inset Formula $p$
|
||
\end_inset
|
||
|
||
-th scatterer, so that the truncated version of the matrix
|
||
\begin_inset Formula $\left(I-T\trops\right)$
|
||
\end_inset
|
||
|
||
is a square matrix with
|
||
\begin_inset Formula $\left(\sum_{p\in\mathcal{P}}N_{p}\right)^{2}$
|
||
\end_inset
|
||
|
||
elements in total.
|
||
The truncated problem
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:Multiple-scattering problem block form"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
can then be solved using standard numerical linear algebra methods (typically,
|
||
by LU factorisation of the
|
||
\begin_inset Formula $\left(I-T\trops\right)$
|
||
\end_inset
|
||
|
||
matrix at a given frequency, and then solving with Gauss elimination for
|
||
as many different incident
|
||
\begin_inset Formula $\rcoeffinc$
|
||
\end_inset
|
||
|
||
vectors as needed).
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Alternatively, the multiple scattering problem can be formulated in terms
|
||
of the regular field expansion coefficients,
|
||
\begin_inset Formula
|
||
\begin{align*}
|
||
\rcoeffp p-\sum_{q\in\mathcal{P}\backslash\left\{ p\right\} }\tropsp pqT_{q}\rcoeffp q & =\rcoeffincp p,\quad p\in\mathcal{P},\\
|
||
\left(I-\trops T\right)\rcoeff & =\rcoeffinc,
|
||
\end{align*}
|
||
|
||
\end_inset
|
||
|
||
but this form is less suitable for numerical calculations due to the fact
|
||
that the regular VSWF expansion coefficients on both sides of the equation
|
||
are typically non-negligible even for large multipole degree
|
||
\begin_inset Formula $l$
|
||
\end_inset
|
||
|
||
, hence the truncation is not justified in this case.
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
TODO less bulshit.
|
||
\end_layout
|
||
|
||
\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 (TODO CHECK CAREFULLY FOR POSSIBLE
|
||
\begin_inset Formula $(-1)^{m'}$
|
||
\end_inset
|
||
|
||
AND SIMILAR FACTORS AND REWRITE IN TERMS OF SPHERICAL HARMONICS)
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Teďka jsou tam zkopírovány výrazy pro C a D z Kristenssona, chybějí fase
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{multline}
|
||
\tropr_{\tau lm;\tau l'm'}\left(\vect d\right)=\frac{\left(-1\right)^{m+m'}}{2}\sum_{\lambda=\left|l-l'\right|}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\
|
||
\times\begin{pmatrix}l & l' & \lambda\\
|
||
0 & 0 & 0
|
||
\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
|
||
m & -m' & m'-m
|
||
\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\
|
||
\times j_{\lambda}\left(d\right)P_{\lambda}^{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}},\\
|
||
\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right)=-i\frac{\left(-1\right)^{m+m'}}{2}\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\
|
||
\times\begin{pmatrix}l & l' & \lambda-1\\
|
||
0 & 0 & 0
|
||
\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
|
||
m & -m' & m'-m
|
||
\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\
|
||
\times j_{\lambda}\left(d\right)P_{\lambda}^{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}},\qquad\tau\ne\tau'.\label{eq:translation operator}
|
||
\end{multline}
|
||
|
||
\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 eqref
|
||
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 eqref
|
||
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 eqref
|
||
reference "eq:regular translation unitarity"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
and composition
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
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 eqref
|
||
reference "eq:atf form multiparticle"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:f squared form multiparticle"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
into
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:scattering CS single"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
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 eqref
|
||
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 eqref
|
||
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 eqref
|
||
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 eqref
|
||
reference "eq:absorption CS multi"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
, but using
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
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 eqref
|
||
reference "eq:absorption CS multi"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:absorption CS multi alternative"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
are equal.
|
||
\end_layout
|
||
|
||
\end_body
|
||
\end_document
|