304 lines
6.0 KiB
Plaintext
304 lines
6.0 KiB
Plaintext
#LyX 2.3 created this file. For more info see http://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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\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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\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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\lang english
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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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\lang english
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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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\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 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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However,
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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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, 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 $\left(\nabla^{2}+k^{2}\right)\Psi\left(\vect r,\vect{\omega}\right)=0$
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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
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\begin_inset Formula $k=TODO$
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\end_inset
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[TODO REF Jackson?].
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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 Standard
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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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literal "true"
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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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\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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\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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