804 lines
16 KiB
Plaintext
804 lines
16 KiB
Plaintext
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\pdf_title "Sähköpajan päiväkirja"
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\pdf_author "Marek Nečada"
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\end_layout
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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 the
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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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That means, among other things, that the
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\begin_inset Formula $T$
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\end_inset
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-matrix is dimensionless and the expansion coefficients of spherical waves
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have units of
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\begin_inset Formula $\sqrt{\mbox{power}}$
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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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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)\label{eq:Absorption is never negative with T}\\
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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 ensured to be true 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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\begin_layout Standard
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Let me rewrite the expression
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Absorption is never negative with T"
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\end_inset
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(without any assumptions about the values of the coefficients
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\begin_inset Formula $a_{n}$
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\end_inset
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) in Dirac notation where the ket
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\begin_inset Formula $\ket a$
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\end_inset
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is the vector of all the exciting wave coefficients
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\begin_inset Formula $a_{n}$
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\end_inset
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.
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Furthemore,
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\begin_inset Formula $\ket{e_{m}}$
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\end_inset
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is the unit vector containing one for the wave indexed by
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\begin_inset Formula $m$
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\end_inset
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and zeros for the rest, so that
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\begin_inset Formula $T_{mn}=\bra{e_{m}}T\ket{e_{n}}$
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\end_inset
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.
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The general expression
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Absorption is never negative with T"
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\end_inset
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and 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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then reads
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\begin_inset Formula
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\begin{eqnarray}
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P & = & \frac{1}{2}\left(\sum_{n}\left|\bra{e_{n}}T\ket a\right|^{2}+\Re\bra aT\ket a\right)\nonumber \\
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& = & \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 \\
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& = & \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}
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\end{eqnarray}
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\end_inset
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giving the following condition 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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||
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\begin_layout Proposition
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||
A
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\begin_inset Formula $T$
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||
\end_inset
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||
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-matrix
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||
\begin_inset Formula $T$
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||
\end_inset
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||
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is unphysical unless the matrix
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||
\begin_inset Formula
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\begin{equation}
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W\equiv\frac{T^{\dagger}T}{2}+\frac{T+T^{\dagger}}{4}\label{eq:Definition of the power matrix}
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\end{equation}
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\end_inset
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is negative (semi)definite.
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||
\end_layout
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||
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\begin_layout Standard
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Obviously, matrix
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||
\begin_inset Formula $W$
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||
\end_inset
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||
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||
is self-adjoint and it has a clear interpretation given by
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||
\begin_inset CommandInset ref
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LatexCommand eqref
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||
reference "eq:Absorption is never negative in Dirac notation"
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||
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||
\end_inset
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||
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||
– for an exciting field given by its expansion coefficient vector
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||
\begin_inset Formula $\ket a$
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\end_inset
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||
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,
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||
\begin_inset Formula $-P=-\bra aW\ket a$
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||
\end_inset
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||
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is the power absorbed by the scatterer.
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||
\end_layout
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||
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||
\begin_layout Subsection
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||
Lossless scatterer
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||
\end_layout
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||
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||
\begin_layout Standard
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Radiation energy conserving scatterer is not very realistic, but it might
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||
provide some simplifications necessary for developing the topological theory.
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||
\end_layout
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||
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||
\begin_layout Standard
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A scatterer always conserves the radiation energy iff
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||
\begin_inset Formula $W=0$
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||
\end_inset
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||
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||
, i.e.
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||
iff
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||
\begin_inset Formula
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||
\[
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||
\frac{T^{\dagger}T}{2}+\frac{T+T^{\dagger}}{4}=0.
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||
\]
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||
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||
\end_inset
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||
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||
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||
\end_layout
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||
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||
\begin_layout Subsubsection
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||
Diagonal
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||
\begin_inset Formula $T$
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||
\end_inset
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||
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||
-matrix
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||
\end_layout
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||
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||
\begin_layout Standard
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||
To get some insight into what does this mean, it might be useful to start
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||
with a diagonal
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||
\begin_inset Formula $T$
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||
\end_inset
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||
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||
-matrix,
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||
\begin_inset Formula $T_{mn}=t_{n}\delta_{mn}$
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||
\end_inset
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||
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||
(valid for e.g.
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||
a spherical particle).
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||
Then for the
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||
\begin_inset Formula $m$
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||
\end_inset
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||
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||
-th matrix element we have
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||
\begin_inset Formula
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||
\[
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||
\left(\Re t_{n}\right)^{2}+\left(\Im t_{n}\right)^{2}+\Re t_{n}=0
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\]
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||
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||
\end_inset
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||
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||
or
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||
\begin_inset Formula
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||
\[
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||
\left(\Re t_{n}+\frac{1}{2}\right)^{2}+\left(\Im t_{n}\right)^{2}=\left(\frac{1}{2}\right)^{2}
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||
\]
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||
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||
\end_inset
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||
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||
which gives a relation between the real and imaginary parts of the scattering
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||
coefficients.
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||
There are two
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||
\begin_inset Quotes eld
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||
\end_inset
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||
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||
extremal
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||
\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
|