hexlaser tmatrixtest begin
Former-commit-id: 447f43d9535318021948783cf086ceb1847b7469
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Marek Nečada
parent
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commit
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#LyX 2.1 created this file. For more info see http://www.lyx.org/
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\lyxformat 474
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\begin_document
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\begin_header
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\textclass article
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\use_default_options true
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\maintain_unincluded_children false
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\language finnish
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\language_package default
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\inputencoding auto
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\fontencoding global
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\font_roman TeX Gyre Pagella
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\font_sans default
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\font_typewriter default
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\font_math auto
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\font_default_family default
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\use_non_tex_fonts true
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\font_sc false
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\font_osf true
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\font_sf_scale 100
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\font_tt_scale 100
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\graphics default
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\default_output_format pdf4
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\output_sync 0
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\bibtex_command default
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\index_command default
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\paperfontsize default
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\spacing single
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\use_hyperref true
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\pdf_title "Sähköpajan päiväkirja"
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\pdf_author "Marek Nečada"
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\pdf_bookmarks true
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\pdf_bookmarksnumbered false
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\pdf_bookmarksopen false
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\pdf_bookmarksopenlevel 1
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\pdf_breaklinks false
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\pdf_pdfborder false
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\pdf_colorlinks false
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\pdf_backref false
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\pdf_pdfusetitle true
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\papersize default
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\use_geometry false
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\use_package amsmath 1
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\use_package amssymb 1
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\use_package cancel 1
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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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\use_package stmaryrd 1
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\use_package undertilde 1
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\cite_engine basic
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\cite_engine_type default
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\biblio_style plain
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\use_bibtopic false
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\use_indices false
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\paperorientation portrait
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\suppress_date false
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\justification true
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\use_refstyle 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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\tocdepth 3
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\paragraph_separation indent
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\paragraph_indentation default
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\quotes_language swedish
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\papercolumns 1
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\papersides 1
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\paperpagestyle default
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\tracking_changes false
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\output_changes false
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\html_math_output 0
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\html_css_as_file 0
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\html_be_strict false
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\end_header
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\begin_body
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\begin_layout Standard
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\lang english
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\begin_inset FormulaMacro
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\newcommand{\vect}[1]{\mathbf{#1}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\ush}[2]{Y_{#1,#2}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\svwfr}[3]{\mathbf{u}_{#1,#2}^{#3}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\svwfs}[3]{\mathbf{v}_{#1,#2}^{#3}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\coeffsi}[3]{a_{#1,#2}^{#3}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\coeffsip}[4]{a_{#1}^{#2,#3,#4}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\coeffri}[3]{p_{#1,#2}^{#3}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\coeffrip}[4]{p_{#1}^{#2,#3,#4}}
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\end_inset
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\end_layout
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\begin_layout Standard
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In this approach, scattering properties of single nanoparticles are first
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computed in terms of vector sperical wavefunctions (VSWFs)—the field incident
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onto the
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\begin_inset Formula $n$
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\end_inset
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-th nanoparticle from external sources can be expanded as
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\begin_inset Formula
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\[
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\vect E_{n}^{\mathrm{inc}}(\vect r)=\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{t=\mathrm{E},\mathrm{M}}\coeffrip nlmt\svwfr lmt\left(\vect r_{n}\right)
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\]
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\end_inset
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where
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\begin_inset Formula $\vect r_{n}=\vect r-\vect R_{n}$
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\end_inset
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,
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\begin_inset Formula $\vect R_{n}$
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\end_inset
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being the position of the centre of
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\begin_inset Formula $n$
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\end_inset
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-th nanoparticle and
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\begin_inset Formula $\svwfr lmt$
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\end_inset
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are the regular VSWFs which can be expressed in terms of regular spherical
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Bessel functions of
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\begin_inset Formula $j_{k}\left(\left|\vect r_{n}\right|\right)$
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\end_inset
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and spherical harmonics
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\begin_inset Formula $\ush km\left(\hat{\vect r}_{n}\right)$
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\end_inset
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; the expressions can be found e.g.
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in REF
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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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few words about different conventions?
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\end_layout
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\end_inset
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.
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On the other hand, the field scattered by the particle can be expanded
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in terms of singular VSWFs
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\begin_inset Formula $\svwfs lmt$
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\end_inset
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which differ from the regular ones by regular spherical Bessel functions
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being replaced with spherical Hankel functions
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\begin_inset Formula $h_{k}^{(1)}\left(\left|\vect r_{n}\right|\right)$
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\end_inset
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,
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\begin_inset Formula
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\[
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\vect E_{n}^{\mathrm{scat}}=\sum_{l,m,t}\coeffsip nlmt\svwfs lmt\left(\vect r_{n}\right).
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\]
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\end_inset
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The expansion coefficients
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\begin_inset Formula $\coeffsip nlmt$
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\end_inset
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,
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\begin_inset Formula $t=\mathrm{E},\mathrm{M}$
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\end_inset
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are related to the electric and magnetic multipole polarisation amplitudes
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of the nanoparticle.
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\end_layout
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\begin_layout Standard
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At a given frequency, assuming the system is linear, the relation between
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the expansion coefficients in the VSWF bases is given by the so-called
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\begin_inset Formula $T$
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\end_inset
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-matrix,
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\begin_inset Formula
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\[
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\coeffsip nlmt=\sum_{l,m,t}T_{n}^{l,m,t;l',m',t'}\coeffrip n{l'}{m'}{t'}.
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\]
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\end_inset
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The
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\begin_inset Formula $T$
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\end_inset
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-matrix is given by the shape and composition of the particle and fully
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describes its scattering properties.
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In theory it is infinite-dimensional, but in practice (at least for subwaveleng
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th nanoparticles) its elements drop very quickly to negligible values with
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growing degree indices
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\begin_inset Formula $l,l'$
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\end_inset
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, enabling to take into account only the elements up to some finite degree,
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\begin_inset Formula $l,l'\le l_{\mathrm{max}}$
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\end_inset
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.
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\end_layout
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\end_body
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\end_document
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