970 lines
23 KiB
Plaintext
970 lines
23 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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\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 Itemize
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motivation (classes of problems that this can solve: response to external
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radiation, resonances, ...)
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\begin_inset Separator latexpar
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\end_inset
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\end_layout
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\begin_deeper
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\begin_layout Itemize
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theory
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\begin_inset Separator latexpar
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\end_inset
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\end_layout
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\begin_deeper
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\begin_layout Itemize
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T-matrix definition, basics
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\begin_inset Separator latexpar
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\end_inset
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\end_layout
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\begin_deeper
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\begin_layout Itemize
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How to get it?
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\end_layout
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\end_deeper
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\begin_layout Itemize
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translation operators (TODO think about how explicit this should be, but
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I guess it might be useful to write them to write them explicitly (but
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in the shortest possible form) in the normalisation used in my program)
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\end_layout
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\begin_layout Itemize
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employing point group symmetries and decomposing the problem to decrease
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the computational complexity (maybe separately)
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\end_layout
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\end_deeper
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\end_deeper
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\begin_layout Subsection
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Motivation
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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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However, the expressions appearing in the re-expansions are fairly complicated,
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and the implementation of MSTMM is extremely error-prone also due to the
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various 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 [REF Jackson?].
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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),\\
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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),
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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),\\
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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),\\
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& \tau=1,2;\quad l=1,2,3,\dots;\quad m=-l,-l+1,\dots,+l,
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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 & =\\
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\vsh 2lm & =\\
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\vsh 3lm & =
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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 [Kristensson 2014];
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over other conventions used elsewhere in literature, it has several fundamental
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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 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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||
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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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||
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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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||
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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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||
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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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||
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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 $\openball 0R\backslash B_{0}\left(R_{<}\right)$
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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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||
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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+\outcoefftlm{\tau}lm\vswfouttlm{\tau}lm\right).\label{eq:E field expansion}
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||
\end{equation}
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||
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||
\end_inset
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||
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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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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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||
.
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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
|
||
|
||
where the
|
||
\begin_inset Formula $T_{\tau lm}^{\tau'l'm'}=T_{\tau lm}^{\tau'l'm'}\left(\omega\right)$
|
||
\end_inset
|
||
|
||
are the elements of the
|
||
\emph on
|
||
transition matrix,
|
||
\emph default
|
||
a.k.a.
|
||
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
-matrix.
|
||
It completely describes the scattering properties of a linear scatterer,
|
||
so with the knowledge of the
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
-matrix, we can solve the single-patricle scatering prroblem simply by substitut
|
||
ing appropriate expansion coefficients
|
||
\begin_inset Formula $\rcoefftlm{\tau'}{l'}{m'}$
|
||
\end_inset
|
||
|
||
of the driving field into
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:T-matrix definition"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset 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 [Kristensson 2016, Mie], but in general one
|
||
can find them numerically by simulating scattering of a regular spherical
|
||
wave
|
||
\begin_inset Formula $\vswfouttlm{\tau}lm$
|
||
\end_inset
|
||
|
||
and projecting the scattered fields (or induced currents, depending on
|
||
the method) onto the outgoing VSWFs
|
||
\begin_inset Formula $\vswfrtlm{\tau}{'l'}{m'}$
|
||
\end_inset
|
||
|
||
.
|
||
In practice, one can compute only a finite number of elements with a cut-off
|
||
value
|
||
\begin_inset Formula $L$
|
||
\end_inset
|
||
|
||
on the multipole degree,
|
||
\begin_inset Formula $l,l'\le L$
|
||
\end_inset
|
||
|
||
, see below.
|
||
We typically use the scuff-tmatrix tool from the free software SCUFF-EM
|
||
suite [SCUFF-EM].
|
||
Note that older versions of SCUFF-EM contained a bug that rendered almost
|
||
all
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
-matrix results wrong; we found and fixed the bug and from upstream version
|
||
xxx onwards, it should behave correctly.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
T-matrix compactness, cutoff validity
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The magnitude of the
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
-matrix elements depends heavily on the scatterer's size compared to the
|
||
wavelength.
|
||
Fortunately, the
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
-matrix of a bounded scatterer is a compact operator [REF???], so from certain
|
||
multipole degree onwards,
|
||
\begin_inset Formula $l,l'>L$
|
||
\end_inset
|
||
|
||
, the elements of the
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
-matrix are negligible, so truncating the
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
-matrix at finite multipole degree
|
||
\begin_inset Formula $L$
|
||
\end_inset
|
||
|
||
gives a good approximation of the actual infinite-dimensional itself.
|
||
If the incident field is well-behaved, i.e.
|
||
the expansion coefficients
|
||
\begin_inset Formula $\rcoefftlm{\tau'}{l'}{m'}$
|
||
\end_inset
|
||
|
||
do not take excessive values for
|
||
\begin_inset Formula $l'>L$
|
||
\end_inset
|
||
|
||
, the scattered field expansion coefficients
|
||
\begin_inset Formula $\outcoefftlm{\tau}lm$
|
||
\end_inset
|
||
|
||
with
|
||
\begin_inset Formula $l>L$
|
||
\end_inset
|
||
|
||
will also be negligible.
|
||
\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 ref
|
||
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 $\openball 0R\backslash B_{0}\left(R_{<}\right)$
|
||
\end_inset
|
||
|
||
have expansion as in
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
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 $\openball 0R\backslash B_{0}\left(R_{<}\right)$
|
||
\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 [REF KRIS] 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 Subsection
|
||
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
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Translation operator
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Numerical complexity, comparison to other methods
|
||
\end_layout
|
||
|
||
\end_body
|
||
\end_document
|