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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 Subsection
Open questions
\end_layout

\begin_layout Subsubsection
How much does the sph.
 harm.
 degree cutoff affect the eigenvalues of 
\begin_inset Formula $W$
\end_inset

?
\end_layout

\begin_layout Standard
When I simulated a cylindrical nanoparticle in scuff-tmatrix (
\begin_inset Formula $l_{\mathrm{max}}=2$
\end_inset

, 50 nm height, 50 nm radius, Palik Ag permittivity) and then with the same
 parameters, just with the imaginary part of permittivity set to zero (i.e.
 without losses), I got almost the same results, including very similar
 eigenvalues of 
\begin_inset Formula $W$
\end_inset

 (although it should then be basically zero).
 This is probably a problem of the BEM method, but it could also be consequence
 of the cutoff.
\end_layout

\begin_layout Standard
For comparison, when I tried exact Mie results for a sphere with 
\begin_inset Formula $\Im\epsilon=0$
\end_inset

, I got also 
\begin_inset Formula $W=0$
\end_inset

 (as expected).
 But 
\begin_inset Formula $T$
\end_inset

-matrix of a sphere is diagonal, hence the cutoff does not affect the eigenvalue
s of resulting (also diagonal) 
\begin_inset Formula $W$
\end_inset

-matrix (below the cutoff, of course).
\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
