Scattering reference – power transport
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Marek Nečada
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@ -108,18 +108,82 @@ Zillion conventions for spherical vector waves
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Legendre polynomials and spherical harmonics: messy from the very beginning
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\end_layout
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\begin_layout Subsection
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Kristensson
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\begin_layout Standard
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\begin_inset Marginal
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status open
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\begin_layout Plain Layout
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FIXME check the Condon-Shortley phases.
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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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Associated Legendre polynomial of degree
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\begin_inset Formula $l\ge0$
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\end_inset
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and order
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\begin_inset Formula $m,$
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\end_inset
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\begin_inset Formula $l\ge m\ge-l$
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\end_inset
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, is given by the recursive relation
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\begin_inset Formula
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\[
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P_{l}^{-m}=\underbrace{\left(-1\right)^{m}}_{\mbox{Condon-Shortley phase}}\frac{1}{2^{l}l!}\left(1-x^{2}\right)^{m/2}\frac{\ud^{l+m}}{\ud x^{l+m}}\left(x^{2}-1\right)^{l}.
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\]
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\end_inset
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There is a relation between the positive and negative orders,
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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P_{l}^{-m}=\left(-1\right)^{m}\frac{\left(l-m\right)!}{\left(l+m\right)!}P_{l}^{m}\left(\cos\theta\right),\quad m\ge0
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P_{l}^{-m}=\underbrace{\left(-1\right)^{m}}_{\mbox{C.-S. p.}}\frac{\left(l-m\right)!}{\left(l+m\right)!}P_{l}^{m}\left(\cos\theta\right),\quad m\ge0.
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\]
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\end_inset
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The index
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\begin_inset Formula $l$
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\end_inset
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(in certain notations, it is often
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\begin_inset Formula $n$
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\end_inset
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) is called
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\emph on
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degree
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\emph default
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, index
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\begin_inset Formula $m$
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\end_inset
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is the
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\emph on
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order
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\emph default
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.
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These two terms are then transitively used for all the object which build
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on the associated Legendre polynomials, i.e.
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spherical harmonics, vector spherical harmonics, spherical waves etc.
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\end_layout
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\begin_layout Subsection
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Kristensson
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\end_layout
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\begin_layout Standard
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Kristensson uses the Condon-Shortley phase, so (sect.
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[K]D.2)
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\end_layout
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@ -528,10 +592,11 @@ In this section I summarize the formulae for power
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\begin_inset Formula $E_{0}$
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\end_inset
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, this can be used to calculate the absorption cross section,
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, this can be used to calculate the absorption cross section (TODO check
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if it should be multiplied by the 2),
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\begin_inset Formula
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\[
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\sigma_{\mathrm{abs}}=-\frac{P}{\varepsilon\varepsilon_{0}\left|E_{0}\right|^{2}}.
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\sigma_{\mathrm{abs}}=-\frac{2P}{\varepsilon\varepsilon_{0}\left|E_{0}\right|^{2}}.
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\]
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\end_inset
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@ -576,7 +641,18 @@ Here
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The radiated power is then (2.28):
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\begin_inset Formula
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\[
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P=\frac{1}{2}\sum_{n}\left(\left|f_{n}\right|^{2}+\Re\left(f_{n}a_{n}^{*}\right)\right)
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P=\frac{1}{2}\sum_{n}\left(\left|f_{n}\right|^{2}+\Re\left(f_{n}a_{n}^{*}\right)\right).
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\]
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\end_inset
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The first term is obviously the power radiated away by the outgoing waves.
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The second term must then be minus the power sucked by the scatterer from
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the exciting wave.
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If the exciting wave is plane, it gives us the extinction cross section
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\begin_inset Formula
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\[
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\sigma_{\mathrm{tot}}=-\frac{\sum_{n}\Re\left(f_{n}a_{n}^{*}\right)}{\varepsilon\varepsilon_{0}\left|E_{0}\right|^{2}}
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\]
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\end_inset
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@ -656,7 +732,15 @@ reference "eq:power-Kristensson-E"
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The radiated power is then
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\begin_inset Formula
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\[
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P=\frac{1}{2}\sum_{m,n}\frac{n\left(n+1\right)}{k^{2}\eta_{0}}\left(\left|a_{mn}\right|^{2}+\left|b_{mn}\right|^{2}-\Re\left(a_{mn}p_{mn}^{*}\right)-\Re\left(b_{mn}q_{mn}^{*}\right)\right).
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P=\frac{1}{2}\sum_{m,n}\frac{n\left(n+1\right)}{k^{2}\eta_{0}\eta}\left(\left|a_{mn}\right|^{2}+\left|b_{mn}\right|^{2}-\Re\left(a_{mn}p_{mn}^{*}\right)-\Re\left(b_{mn}q_{mn}^{*}\right)\right).
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\]
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\end_inset
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If the exciting wave is a plane wave, the extinction cross section is
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\begin_inset Formula
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\[
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\sigma_{\mathrm{tot}}=\frac{\Re\left(a_{mn}p_{mn}^{*}\right)+\Re\left(b_{mn}q_{mn}^{*}\right)}{\varepsilon\varepsilon_{0}\left|E_{0}\right|^{2}}
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\]
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\end_inset
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@ -689,15 +773,166 @@ Near field limit
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\end_layout
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\begin_layout Chapter
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Mie Theory
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Single particle scattering and Mie theory
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\end_layout
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\begin_layout Standard
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The basic idea is simple.
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For an exciting spherical wave (usually the regular wave in whatever convention
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) of a given frequency
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\begin_inset Formula $\omega$
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\end_inset
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, type
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\begin_inset Formula $\zeta$
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\end_inset
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(electric or magnetic), degree
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\begin_inset Formula $l$
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\end_inset
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and order
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\begin_inset Formula $m$
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\end_inset
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, the particle responds with waves from the complementary set (e.g.
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outgoing waves), with the same frequency
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\begin_inset Formula $\omega$
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\end_inset
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, but any type
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\begin_inset Formula $\zeta'$
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\end_inset
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, degree
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\begin_inset Formula $l'$
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\end_inset
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and order
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\begin_inset Formula $m'$
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\end_inset
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, in a way that the Maxwell's equations are satisfied, with the coefficients
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\begin_inset Formula $T_{l',m';l,m}^{\zeta',\zeta}(\omega)$
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\end_inset
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.
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This yields one row in the scattering matrix (often called the
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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 $T(\omega)$
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\end_inset
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, which fully characterizes the scattering properties of the particle (in
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the linear regime, of course).
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Analytical expression for the matrix is known for spherical scatterer,
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otherwise it is computed numerically (using DDA, BEM or whatever).
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So if we have the two sets of spherical wave functions
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\begin_inset Formula $\vect f_{lm}^{J_{1},\zeta}$
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\end_inset
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,
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\begin_inset Formula $\vect f_{lm}^{J_{2},\zeta}$
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\end_inset
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and the full
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\begin_inset Quotes sld
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\end_inset
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exciting
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\begin_inset Quotes srd
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\end_inset
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wave has electric field given as
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\begin_inset Formula
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\[
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\vect E_{\mathrm{ext}}=\sum_{\zeta=\mathrm{E,M}}\sum_{l,m}c_{lm}^{\zeta}\vect f_{lm}^{\zeta},
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\]
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\end_inset
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the
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\begin_inset Quotes sld
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\end_inset
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scattered
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\begin_inset Quotes srd
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\end_inset
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field will be
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\begin_inset Formula
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\[
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\vect E_{\mathrm{scat}}=\sum_{\zeta',l',m'}\sum_{\zeta,l,m}T_{l',m';l,m}^{\zeta',\zeta}c_{lm}^{\zeta}\vect f_{l'm'}^{\zeta'},
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\]
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\end_inset
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and the total field around the scaterer is
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\begin_inset Formula $\vect E=\vect E_{\mathrm{ext}}+\vect E_{\mathrm{scat}}$
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\end_inset
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.
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\end_layout
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\begin_layout Section
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Full version
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Mie theory – full version
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\end_layout
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\begin_layout Standard
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\begin_inset Formula $T$
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\end_inset
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-matrix for a spherical particle is type-, degree- and order- diagonal,
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that is,
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\begin_inset Formula $T_{l',m';l,m}^{\zeta',\zeta}(\omega)=0$
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\end_inset
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if
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\begin_inset Formula $l\ne l'$
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\end_inset
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,
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\begin_inset Formula $m\ne m'$
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\end_inset
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or
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\begin_inset Formula $\zeta\ne\zeta'$
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\end_inset
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.
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Moreover, it does not depend on
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\begin_inset Formula $m$
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\end_inset
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, so
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\begin_inset Formula
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\[
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T_{l',m';l,m}^{\zeta',\zeta}(\omega)=T_{l}^{\zeta}\left(\omega\right)\delta_{\zeta'\zeta}\delta_{l'l}\delta_{m'm}
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\]
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\end_inset
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where for the usual choice
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\begin_inset Formula $J_{1}=1,J_{2}=3$
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\end_inset
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\begin_inset Formula
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\begin{eqnarray*}
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T_{l}^{E}\left(\omega\right) & = & TODO,\\
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T_{l}^{M}(\omega) & = & TODO.
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\end{eqnarray*}
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\end_inset
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\end_layout
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\begin_layout Section
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Long wave approximation
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Long wave approximation for spherical nanoparticle
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\end_layout
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\begin_layout Standard
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