Begin memo on radiative xfer.
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Marek Nečada
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\begin_layout Title
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Radiation power balance in nanoparticles
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\end_layout
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\begin_layout Author
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Marek Nečada
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\end_layout
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\begin_layout Abstract
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This memo deals with the formulae for radiation transfer, absorption, extinction
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for single particle and composite system of several nanoparticles.
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I also derive some natural conditions on
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\begin_inset Formula $T$
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\end_inset
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-matrix elements.
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\end_layout
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\begin_layout Section*
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Conventions
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\end_layout
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\begin_layout Standard
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If not stated otherwise, Kristensson's notation and normalisation conventions
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are used in this memo.
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\end_layout
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\begin_layout Section
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Single particle
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\end_layout
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\begin_layout Subsection
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Power transfer formula, absorption
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\end_layout
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\begin_layout Standard
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The power radiated away by a linear scatterer at fixed harmonic frequency
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is according to [Kris (2.28)]
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\begin_inset Formula
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\[
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P=\frac{1}{2}\sum_{n}\left(\left|f_{n}\right|^{2}+\Re\left(f_{n}a_{n}^{*}\right)\right)
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\]
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\end_inset
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where
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\begin_inset Formula $n$
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\end_inset
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is a multiindex describing the type (E/M) and multipole degree and order
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of the wave,
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\begin_inset Formula $f_{n}$
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\end_inset
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is the coefficient corresponding to
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\series bold
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outgoing
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\series default
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(Hankel function based) and
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\begin_inset Formula $a_{n}$
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\end_inset
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to
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\series bold
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regular
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\series default
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(first-order Bessel function based) waves.
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\end_layout
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\begin_layout Standard
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This is minus the power absorbed by the nanoparticle, and unless the particle
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has some gain mechanism, this cannot be positive.
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The basic condition for a physical nanoparticle therefore reads
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\begin_inset Formula
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\begin{equation}
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P=\frac{1}{2}\sum_{n}\left(\left|f_{n}\right|^{2}+\Re\left(f_{n}a_{n}^{*}\right)\right)\le0.\label{eq:Absorption is never negative}
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Subsection
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Conditions on the
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\begin_inset Formula $T$
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\end_inset
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-matrix
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\end_layout
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\begin_layout Standard
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For a linear scatterer, the outgoing and regular wave coefficients are connected
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via 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
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\begin{equation}
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f_{n}=\sum_{n'}T_{nn'}a_{n'}.\label{eq:T-matrix definition}
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Standard
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Inequality
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Absorption is never negative"
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\end_inset
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enables us to derive some conditions on the
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\begin_inset Formula $T$
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\end_inset
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-matrix.
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Let the particle be driven by a wave of a single type
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\begin_inset Formula $m$
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\end_inset
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only so the coefficients of all other components of the driving field are
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zero,
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\begin_inset Formula $a_{n}=\delta_{nm}$
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\end_inset
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.
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From
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Absorption is never negative"
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\end_inset
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and
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:T-matrix definition"
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\end_inset
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we get
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\begin_inset Formula
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\begin{eqnarray}
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P & = & \frac{1}{2}\sum_{n}\left(\left|\sum_{n'}T_{nn'}a_{n'}\right|^{2}+\Re\left(\sum_{n'}T_{nn'}a_{n'}a_{n}^{*}\right)\right)\nonumber \\
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& = & \frac{1}{2}\sum_{n}\left(\left|\sum_{n'}T_{nn'}\delta_{n'm}\right|^{2}+\Re\left(\sum_{n'}T_{nn'}\delta_{n'm}\delta_{nm}\right)\right)\nonumber \\
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& = & \frac{1}{2}\left(\left|\sum_{n}T_{nm}\right|^{2}+\Re T_{mm}\right)\le0\qquad\forall m,\label{eq:Absorption is never negative for single wave type}
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\end{eqnarray}
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\end_inset
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a condition that should be checked e.g.
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for the
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\begin_inset Formula $T$
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\end_inset
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-matrices generated by SCUFF-EM.
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\end_layout
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\begin_layout Remark
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For a particle of spherical symmetry
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\begin_inset Formula $T_{nm}\propto\delta_{nm}$
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\end_inset
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, so
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Absorption is never negative for single wave type"
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\end_inset
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gives
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\begin_inset Formula $-\Re T_{mm}\ge\left|T_{mm}\right|^{2}$
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\end_inset
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which in turn implies
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\begin_inset Formula $\left|T_{mm}\right|<1$
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\end_inset
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.
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(Any similar conclusion for the general case?)
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\end_layout
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\begin_layout Problem
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Obviously,
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Absorption is never negative for single wave type"
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\end_inset
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is the consequence of the condition
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Absorption is never negative"
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\end_inset
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.
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But is
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Absorption is never negative"
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\end_inset
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always true if
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Absorption is never negative for single wave type"
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\end_inset
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satisfied?
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\end_layout
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\end_body
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\end_document
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