qpms/notes/radpower.lyx

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\pdf_title "Sähköpajan päiväkirja"
\pdf_author "Marek Nečada"
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\begin_layout Title
Radiation power balance in nanoparticles
\end_layout

\begin_layout Author
Marek Nečada
\end_layout

\begin_layout Abstract
This memo deals with the formulae for radiation transfer, absorption, extinction
 for single particle and composite system of several nanoparticles.
 I also derive some natural conditions on 
\begin_inset Formula $T$
\end_inset

-matrix elements.
\end_layout

\begin_layout Section*
Conventions
\end_layout

\begin_layout Standard
If not stated otherwise, Kristensson's notation and normalisation conventions
 are used in this memo.
\end_layout

\begin_layout Section
Single particle
\end_layout

\begin_layout Subsection
Power transfer formula, absorption
\end_layout

\begin_layout Standard
The power radiated away by a linear scatterer at fixed harmonic frequency
 is according to [Kris (2.28)]
\begin_inset Formula 
\[
P=\frac{1}{2}\sum_{n}\left(\left|f_{n}\right|^{2}+\Re\left(f_{n}a_{n}^{*}\right)\right)
\]

\end_inset

where 
\begin_inset Formula $n$
\end_inset

 is a multiindex describing the type (E/M) and multipole degree and order
 of the wave, 
\begin_inset Formula $f_{n}$
\end_inset

 is the coefficient corresponding to 
\series bold
outgoing
\series default
 (Hankel function based) and 
\begin_inset Formula $a_{n}$
\end_inset

 to 
\series bold
regular
\series default
 (first-order Bessel function based) waves.
\end_layout

\begin_layout Standard
This is minus the power absorbed by the nanoparticle, and unless the particle
 has some gain mechanism, this cannot be positive.
 The basic condition for a physical nanoparticle therefore reads
\begin_inset Formula 
\begin{equation}
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}
\end{equation}

\end_inset


\end_layout

\begin_layout Subsection
Conditions on the 
\begin_inset Formula $T$
\end_inset

-matrix
\end_layout

\begin_layout Standard
For a linear scatterer, the outgoing and regular wave coefficients are connected
 via the 
\begin_inset Formula $T$
\end_inset

-matrix
\begin_inset Formula 
\begin{equation}
f_{n}=\sum_{n'}T_{nn'}a_{n'}.\label{eq:T-matrix definition}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Inequality 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Absorption is never negative"

\end_inset

 enables us to derive some conditions on the 
\begin_inset Formula $T$
\end_inset

-matrix.
 Let the particle be driven by a wave of a single type 
\begin_inset Formula $m$
\end_inset

 only so the coefficients of all other components of the driving field are
 zero, 
\begin_inset Formula $a_{n}=\delta_{nm}$
\end_inset

.
 From 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Absorption is never negative"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:T-matrix definition"

\end_inset

 we get
\begin_inset Formula 
\begin{eqnarray}
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 \\
 & = & \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 \\
 & = & \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}
\end{eqnarray}

\end_inset

a condition that should be checked e.g.
 for the 
\begin_inset Formula $T$
\end_inset

-matrices generated by SCUFF-EM.
\end_layout

\begin_layout Remark
For a particle of spherical symmetry 
\begin_inset Formula $T_{nm}\propto\delta_{nm}$
\end_inset

, so 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Absorption is never negative for single wave type"

\end_inset

 gives 
\begin_inset Formula $-\Re T_{mm}\ge\left|T_{mm}\right|^{2}$
\end_inset

 which in turn implies 
\begin_inset Formula $\left|T_{mm}\right|<1$
\end_inset

.
 (Any similar conclusion for the general case?)
\end_layout

\begin_layout Problem
Obviously, 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Absorption is never negative for single wave type"

\end_inset

 is the consequence of the condition 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Absorption is never negative"

\end_inset

.
 But is 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Absorption is never negative"

\end_inset

 always true if 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Absorption is never negative for single wave type"

\end_inset

 satisfied?
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