#LyX 2.1 created this file. For more info see http://www.lyx.org/ \lyxformat 474 \begin_document \begin_header \textclass article \use_default_options true \begin_modules theorems-ams \end_modules \maintain_unincluded_children false \language english \language_package default \inputencoding auto \fontencoding global \font_roman TeX Gyre Pagella \font_sans default \font_typewriter default \font_math auto \font_default_family default \use_non_tex_fonts true \font_sc false \font_osf true \font_sf_scale 100 \font_tt_scale 100 \graphics default \default_output_format pdf4 \output_sync 0 \bibtex_command default \index_command default \paperfontsize 10 \spacing single \use_hyperref true \pdf_title "Sähköpajan päiväkirja" \pdf_author "Marek Nečada" \pdf_bookmarks true \pdf_bookmarksnumbered false \pdf_bookmarksopen false \pdf_bookmarksopenlevel 1 \pdf_breaklinks false \pdf_pdfborder false \pdf_colorlinks false \pdf_backref false \pdf_pdfusetitle true \papersize a4paper \use_geometry true \use_package amsmath 1 \use_package amssymb 1 \use_package cancel 1 \use_package esint 1 \use_package mathdots 1 \use_package mathtools 1 \use_package mhchem 1 \use_package stackrel 1 \use_package stmaryrd 1 \use_package undertilde 1 \cite_engine basic \cite_engine_type default \biblio_style plain \use_bibtopic false \use_indices false \paperorientation portrait \suppress_date false \justification true \use_refstyle 1 \index Index \shortcut idx \color #008000 \end_index \leftmargin 1cm \topmargin 5mm \rightmargin 1cm \bottommargin 1cm \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \paragraph_indentation default \quotes_language english \papercolumns 1 \papersides 1 \paperpagestyle default \tracking_changes false \output_changes false \html_math_output 0 \html_css_as_file 0 \html_be_strict false \end_header \begin_body \begin_layout Standard \begin_inset FormulaMacro \newcommand{\uoft}[1]{\mathfrak{F}#1} \end_inset \begin_inset FormulaMacro \newcommand{\uaft}[1]{\mathfrak{\mathbb{F}}#1} \end_inset \begin_inset FormulaMacro \newcommand{\usht}[2]{\mathbb{S}_{#1}#2} \end_inset \begin_inset FormulaMacro \newcommand{\bsht}[2]{\mathrm{S}_{#1}#2} \end_inset \begin_inset FormulaMacro \newcommand{\pht}[2]{\mathfrak{\mathbb{H}}_{#1}#2} \end_inset \begin_inset FormulaMacro \newcommand{\vect}[1]{\mathbf{#1}} \end_inset \begin_inset FormulaMacro \newcommand{\ud}{\mathrm{d}} \end_inset \begin_inset FormulaMacro \newcommand{\basis}[1]{\mathfrak{#1}} \end_inset \begin_inset FormulaMacro \newcommand{\dc}[1]{Ш_{#1}} \end_inset \begin_inset FormulaMacro \newcommand{\rec}[1]{#1^{-1}} \end_inset \begin_inset FormulaMacro \newcommand{\recb}[1]{#1^{\widehat{-1}}} \end_inset \begin_inset FormulaMacro \newcommand{\ints}{\mathbb{Z}} \end_inset \begin_inset FormulaMacro \newcommand{\nats}{\mathbb{N}} \end_inset \begin_inset FormulaMacro \newcommand{\reals}{\mathbb{R}} \end_inset \begin_inset FormulaMacro \newcommand{\ush}[2]{Y_{#1,#2}} \end_inset \begin_inset FormulaMacro \newcommand{\hgfr}{\mathbf{F}} \end_inset \begin_inset FormulaMacro \newcommand{\ph}{\mathrm{ph}} \end_inset \begin_inset FormulaMacro \newcommand{\kor}[1]{\underline{#1}} \end_inset \begin_inset FormulaMacro \newcommand{\koru}[1]{\overline{#1}} \end_inset \begin_inset FormulaMacro \newcommand{\hgf}{F} \end_inset \begin_inset FormulaMacro \newcommand{\bra}[1]{\left\langle #1\right|} \end_inset \begin_inset FormulaMacro \newcommand{\ket}[1]{\left|#1\right\rangle } \end_inset \begin_inset FormulaMacro \newcommand{\sci}[1]{\mathfrak{#1}} \end_inset \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