419 lines
8.7 KiB
Plaintext
419 lines
8.7 KiB
Plaintext
#LyX 2.1 created this file. For more info see http://www.lyx.org/
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\pdf_author "Marek Nečada"
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\use_package esint 1
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\use_package mathdots 1
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\use_package mathtools 1
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\use_package mhchem 1
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\use_package stackrel 1
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\index Index
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\shortcut idx
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\color #008000
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\end_index
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\secnumdepth 3
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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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\begin_inset CommandInset label
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LatexCommand label
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name "sec:Finite"
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\end_inset
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\end_layout
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\begin_layout Itemize
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\lang english
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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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\end_layout
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\begin_deeper
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\begin_layout Itemize
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\lang english
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theory
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\end_layout
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\begin_deeper
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\begin_layout Itemize
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\lang english
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T-matrix definition, basics
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\end_layout
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\begin_deeper
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\begin_layout Itemize
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\lang english
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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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\lang english
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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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\lang english
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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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\lang english
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Motivation
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\end_layout
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\begin_layout Subsection
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\lang english
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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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depending only on (angular) frequency
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\begin_inset Formula $\omega$
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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 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
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\begin_inset Formula $\left(\nabla^{2}+k^{2}\right)\vect{\Psi}\left(\vect r,\vect{\omega}\right)=0$
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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 wavenumber
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\begin_inset Formula $k=\omega\sqrt{\mu\epsilon}/c_{0}$
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\end_inset
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, and transversality condition
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\begin_inset Formula $\nabla\cdot\vect{\Psi}\left(\vect r,\omega\right)=0$
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\end_inset
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\begin_inset CommandInset citation
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LatexCommand cite
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key "jackson_classical_1998"
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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 more specific REF Jackson?]
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\end_layout
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\end_inset
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.
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\lang english
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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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\begin_layout Plain Layout
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\lang english
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Throughout this text, we will use the same normalisation conventions as
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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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\end_inset
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.
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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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\lang english
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Spherical waves
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\end_layout
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\begin_layout Standard
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Inside a ball
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\begin_inset Formula $B_{R}\left(\vect{r'}\right)\subset\medium$
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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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centered at
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\begin_inset Formula $\vect{r'}$
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\end_inset
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, the transversal solutions of the vector Helmholtz equation can be expressed
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in the basis of the regular transversal
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\emph on
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vector spherical wavefunctions
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\emph default
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(VSWFs)
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\begin_inset Formula $\vswfr{\tau}lm\left(k\left(\vect r-\vect{r'}\right)\right)$
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\end_inset
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, which are found by separation of variables in spherical coordinates.
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There is a large variety of VSWF normalisation and phase conventions in
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the literature (and existing software), which can lead to great confusion
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using them.
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Throughout this text, we use the following convention, adopted from [Kristensso
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n 2014]:
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{eqnarray}
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\vswfr 1lm\left(k\vect r\right) & = & j_{l}\left(kr\right)\vspharm 1lm\left(\uvec r\right),\nonumber \\
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\vswfr 2lm\left(k\vect r\right) & = & \frac{1}{kr}\frac{\ud\left(kr\, j_{l}\left(kr\right)\right)}{\ud\left(kr\right)}\vspharm 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(kr\right)}{kr}\vspharm 3lm\left(\uvec r\right),\label{eq:regular vswf}\\
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& & \qquad l=1,2,\dots;\, m=-l,-l+1,\dots,l;\nonumber
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\end{eqnarray}
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\end_inset
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where we separated the position variable into its magnitude
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\begin_inset Formula $r$
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\end_inset
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and a unit vector
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\begin_inset Formula $\uvec r$
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\end_inset
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,
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\begin_inset Formula $\vect r=r\uvec r$
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\end_inset
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, 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 $\vspharm{\sigma}lm$
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\end_inset
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are defined as
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\begin_inset Formula
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\begin{eqnarray}
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\vspharm 1lm\left(\uvec r\right) & = & \frac{1}{\sqrt{l\left(l+1\right)}}\nabla\spharm lm\left(\uvec r\right)\times\vect r,\nonumber \\
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\vspharm 2lm\left(\uvec r\right) & = & \frac{1}{\sqrt{l\left(l+1\right)}}r\nabla\spharm lm\left(\uvec r\right),\label{eq:vspharm}\\
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\vspharm 2lm\left(\uvec r\right) & = & \uvec r\spharm lm\left(\uvec r\right),\nonumber
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\end{eqnarray}
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\end_inset
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and for the scalar spherical harmonics
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\begin_inset Formula $\spharm lm$
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\end_inset
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we use the convention from [REF DLMF 14.30.1],
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\begin_inset Formula
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\begin{equation}
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\spharm lm\left(\uvec r\right)=\spharm lm\left(\theta,\phi\right)=\left(\frac{(l-m)!(2l+1)}{4\pi(l+m)!}\right)^{1/2}e^{im\phi}\mathsf{P}_{l}^{m}\left(\cos\theta\right),\label{eq:spharm}
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\end{equation}
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\end_inset
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where the Condon-Shortley phase factor
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\begin_inset Formula $\left(-1\right)^{m}$
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\end_inset
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is already included in the definition of Ferrers function
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\begin_inset Formula $\mathsf{P}_{l}^{m}\left(\cos\theta\right)$
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\end_inset
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[as in DLMF 14].
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The main reason for this choice of VSWF
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\emph on
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normalisation
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\emph default
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is that it leads to simple formulae for power transport and scattering
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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, see below.
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\end_layout
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\begin_layout Standard
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\lang english
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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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\lang english
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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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\lang english
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T-matrix definition
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\end_layout
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\begin_layout Subsubsection
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Absorbed power
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\end_layout
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\begin_layout Subsubsection
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\lang english
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T-matrix compactness, cutoff validity
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\end_layout
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\begin_layout Subsection
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\lang english
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Multiple scattering
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\end_layout
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\begin_layout Subsubsection
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\lang english
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Translation operator
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\end_layout
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\begin_layout Subsubsection
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\lang english
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Numerical complexity, comparison to other methods
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\end_layout
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\end_body
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\end_document
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