qpms/notes/radpower.lyx

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\pdf_title "Sähköpajan päiväkirja"
\pdf_author "Marek Nečada"
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\begin_body
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\end_layout
\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 the
\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.
That means, among other things, that the
\begin_inset Formula $T$
\end_inset
-matrix is dimensionless and the expansion coefficients of spherical waves
have units of
\begin_inset Formula $\sqrt{\mbox{power}}$
\end_inset
.
\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)\label{eq:Absorption is never negative with T}\\
& = & \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 ensured to be true 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?
\end_layout
\begin_layout Standard
Let me rewrite the expression
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Absorption is never negative with T"
\end_inset
(without any assumptions about the values of the coefficients
\begin_inset Formula $a_{n}$
\end_inset
) in Dirac notation where the ket
\begin_inset Formula $\ket a$
\end_inset
is the vector of all the exciting wave coefficients
\begin_inset Formula $a_{n}$
\end_inset
.
Furthemore,
\begin_inset Formula $\ket{e_{m}}$
\end_inset
is the unit vector containing one for the wave indexed by
\begin_inset Formula $m$
\end_inset
and zeros for the rest, so that
\begin_inset Formula $T_{mn}=\bra{e_{m}}T\ket{e_{n}}$
\end_inset
.
The general expression
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Absorption is never negative with T"
\end_inset
and condition
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Absorption is never negative"
\end_inset
then reads
\begin_inset Formula
\begin{eqnarray}
P & = & \frac{1}{2}\left(\sum_{n}\left|\bra{e_{n}}T\ket a\right|^{2}+\Re\bra aT\ket a\right)\nonumber \\
& = & \frac{1}{2}\left(\sum_{n}\bra aT^{\dagger}\ket{e_{n}}\bra{e_{n}}T\ket a+\frac{1}{2}\left(\bra aT\ket a+\bra aT\ket a^{*}\right)\right)\nonumber \\
& = & \frac{1}{2}\bra aT^{\dagger}T\ket a+\frac{1}{4}\bra a\left(T+T^{\dagger}\right)\ket a\le0\qquad\forall\ket a,\label{eq:Absorption is never negative in Dirac notation}
\end{eqnarray}
\end_inset
giving the following condition on the
\begin_inset Formula $T$
\end_inset
-matrix:
\end_layout
\begin_layout Proposition
A
\begin_inset Formula $T$
\end_inset
-matrix
\begin_inset Formula $T$
\end_inset
is unphysical unless the matrix
\begin_inset Formula
\begin{equation}
W\equiv\frac{T^{\dagger}T}{2}+\frac{T+T^{\dagger}}{4}\label{eq:Definition of the power matrix}
\end{equation}
\end_inset
is negative (semi)definite.
\end_layout
\begin_layout Standard
Obviously, matrix
\begin_inset Formula $W$
\end_inset
is self-adjoint and it has a clear interpretation given by
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Absorption is never negative in Dirac notation"
\end_inset
for an exciting field given by its expansion coefficient vector
\begin_inset Formula $\ket a$
\end_inset
,
\begin_inset Formula $-P=-\bra aW\ket a$
\end_inset
is the power absorbed by the scatterer.
\end_layout
\begin_layout Subsection
Lossless scatterer
\end_layout
\begin_layout Standard
Radiation energy conserving scatterer is not very realistic, but it might
provide some simplifications necessary for developing the topological theory.
\end_layout
\begin_layout Standard
A scatterer always conserves the radiation energy iff
\begin_inset Formula $W=0$
\end_inset
, i.e.
iff
\begin_inset Formula
\[
\frac{T^{\dagger}T}{2}+\frac{T+T^{\dagger}}{4}=0.
\]
\end_inset
\end_layout
\begin_layout Subsubsection
Diagonal
\begin_inset Formula $T$
\end_inset
-matrix
\end_layout
\begin_layout Standard
To get some insight into what does this mean, it might be useful to start
with a diagonal
\begin_inset Formula $T$
\end_inset
-matrix,
\begin_inset Formula $T_{mn}=t_{n}\delta_{mn}$
\end_inset
(valid for e.g.
a spherical particle).
Then for the
\begin_inset Formula $m$
\end_inset
-th matrix element we have
\begin_inset Formula
\[
\left(\Re t_{n}\right)^{2}+\left(\Im t_{n}\right)^{2}+\Re t_{n}=0
\]
\end_inset
or
\begin_inset Formula
\[
\left(\Re t_{n}+\frac{1}{2}\right)^{2}+\left(\Im t_{n}\right)^{2}=\left(\frac{1}{2}\right)^{2}
\]
\end_inset
which gives a relation between the real and imaginary parts of the scattering
coefficients.
There are two
\begin_inset Quotes eld
\end_inset
extremal
\begin_inset Quotes erd
\end_inset
real values,
\begin_inset Formula $t_{n}=0$
\end_inset
(no scattering at all) and
\begin_inset Formula $t_{n}=-1$
\end_inset
.
In general, the possible values lie on a half-unit circle in the complex
plane with the centre at
\begin_inset Formula $-1/2$
\end_inset
.
The half-unit disk delimited by the circle is the (realistic) lossy region,
while everything outside it represents (unrealistic) system with gain.
\end_layout
\begin_layout Section
Multiple scattering
\end_layout
\begin_layout Standard
The purpose of this section is to clarify the formulae for absorption and
extinction in a system of multiple scatterers.
Let the scatterers be indexed by fraktur letters, so the power
\begin_inset Quotes eld
\end_inset
generated
\begin_inset Quotes erd
\end_inset
by nanoparticle
\begin_inset Formula $\sci k$
\end_inset
will be denoted as
\begin_inset Formula $P^{\sci k}$
\end_inset
.
Quantities without such indices apply
\begin_inset Note Note
status open
\begin_layout Plain Layout
se vztahují
\end_layout
\end_inset
to the whole system, so
\begin_inset Formula $P$
\end_inset
will now denote the total power generated by the system.
Now
\begin_inset Formula $\ket{a_{0}^{\sci k}}$
\end_inset
is the expansion of the external driving field in the location of nanoparticle
\begin_inset Formula $\sci k$
\end_inset
and
\begin_inset Formula $\ket{a^{\sci k}}$
\end_inset
is the expansion of the external field together with the fields scattered
from other nanoparticles,
\begin_inset Formula
\[
\ket{a^{\sci k}}=\ket{a_{0}^{\sci k}}+\sum_{\sci l\ne\sci k}S_{\sci k\leftarrow\sci l}\ket{f^{\sci l}}.
\]
\end_inset
Rewriting
\begin_inset Formula $\ket{f^{\sci l}}=T^{\sci l}\ket{a^{\sci l}}$
\end_inset
, this gives the scattering problem in terms of
\begin_inset Formula $\ket{a^{\sci k}}$
\end_inset
,
\begin_inset Formula
\[
\ket{a^{\sci k}}=\ket{a_{0}^{\sci k}}+\sum_{\sci l\ne\sci k}S_{\sci k\leftarrow\sci l}T^{\sci l}\ket{a^{\sci l}}
\]
\end_inset
or, in the indexless notation for the whole system
\begin_inset Formula
\begin{eqnarray*}
\ket a & = & \ket{a_{0}}+ST\ket a,\\
\left(1-ST\right)\ket a & = & \ket{a_{0}}
\end{eqnarray*}
\end_inset
Alternatively, multiplication by
\begin_inset Formula $T$
\end_inset
from the left gives the problem in terms of the outgoing wave coefficients,
\begin_inset Formula
\begin{eqnarray*}
\ket f & = & T\ket{a_{0}}+TS\ket f,\\
\left(1-TS\right)\ket f & = & T\ket{a_{0}}.
\end{eqnarray*}
\end_inset
\end_layout
\begin_layout Standard
\series bold
TODO
\end_layout
\end_body
\end_document