#LyX 2.1 created this file. For more info see http://www.lyx.org/ \lyxformat 474 \begin_document \begin_header \textclass report \begin_preamble %\renewcommand*{\chapterheadstartvskip}{\vspace*{1cm}} %\renewcommand*{\chapterheadendvskip}{\vspace{2cm}} \end_preamble \use_default_options true \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 default \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 a5paper \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 2cm \topmargin 2cm \rightmargin 2cm \bottommargin 2cm \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \paragraph_indentation default \quotes_language swedish \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{\vect}[1]{\mathbf{#1}} \end_inset \begin_inset FormulaMacro \newcommand{\ud}{\mathrm{d}} \end_inset \end_layout \begin_layout Title Electromagnetic multiple scattering, spherical waves and **** \end_layout \begin_layout Author Marek Nečada \end_layout \begin_layout Chapter Zillion conventions for spherical vector waves \end_layout \begin_layout Section Legendre polynomials and spherical harmonics: messy from the very beginning \end_layout \begin_layout Standard \begin_inset Marginal status open \begin_layout Plain Layout FIXME check the Condon-Shortley phases. \end_layout \end_inset \end_layout \begin_layout Standard Associated Legendre polynomial of degree \begin_inset Formula $l\ge0$ \end_inset and order \begin_inset Formula $m,$ \end_inset \begin_inset Formula $l\ge m\ge-l$ \end_inset , is given by the recursive relation \begin_inset Formula \[ P_{l}^{-m}=\underbrace{\left(-1\right)^{m}}_{\mbox{Condon-Shortley phase}}\frac{1}{2^{l}l!}\left(1-x^{2}\right)^{m/2}\frac{\ud^{l+m}}{\ud x^{l+m}}\left(x^{2}-1\right)^{l}. \] \end_inset There is a relation between the positive and negative orders, \end_layout \begin_layout Standard \begin_inset Formula \[ P_{l}^{-m}=\underbrace{\left(-1\right)^{m}}_{\mbox{C.-S. p.}}\frac{\left(l-m\right)!}{\left(l+m\right)!}P_{l}^{m}\left(\cos\theta\right),\quad m\ge0. \] \end_inset The index \begin_inset Formula $l$ \end_inset (in certain notations, it is often \begin_inset Formula $n$ \end_inset ) is called \emph on degree \emph default , index \begin_inset Formula $m$ \end_inset is the \emph on order \emph default . These two terms are then transitively used for all the object which build on the associated Legendre polynomials, i.e. spherical harmonics, vector spherical harmonics, spherical waves etc. \end_layout \begin_layout Subsection Kristensson \end_layout \begin_layout Standard Kristensson uses the Condon-Shortley phase, so (sect. [K]D.2) \end_layout \begin_layout Standard \begin_inset Formula \[ Y_{lm}\left(\hat{\vect r}\right)=\left(-1\right)^{m}\sqrt{\frac{2l+1}{4\pi}\frac{\left(l-m\right)!}{\left(l+m\right)!}}P_{l}^{m}\left(\cos\theta\right)e^{im\phi} \] \end_inset \begin_inset Formula \[ Y_{lm}^{\dagger}\left(\hat{\vect r}\right)=Y_{lm}^{*}\left(\hat{\vect r}\right) \] \end_inset \begin_inset Formula \[ Y_{l,-m}\left(\hat{\vect r}\right)=\left(-1\right)^{m}Y_{lm}^{\dagger}\left(\hat{\vect r}\right) \] \end_inset \end_layout \begin_layout Standard Orthonormality: \begin_inset Formula \[ \int Y_{lm}\left(\hat{\vect r}\right)Y_{l'm'}^{\dagger}\left(\hat{\vect r}\right)\,\ud\Omega=\delta_{ll'}\delta_{mm'} \] \end_inset \end_layout \begin_layout Section Pi and tau \end_layout \begin_layout Subsection Xu \begin_inset CommandInset label LatexCommand label name "sub:Xu pitau" \end_inset \end_layout \begin_layout Standard As in (37) \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray*} \pi_{mn}\left(\cos\theta\right) & = & \frac{m}{\sin\theta}P_{n}^{m}\left(\cos\theta\right)\\ \tau_{mn}\left(\cos\theta\right) & = & \frac{\ud}{\ud\theta}P_{n}^{m}\left(\cos\theta\right)=-\left(\sin\theta\right)\frac{\ud P_{n}^{m}\left(\cos\theta\right)}{\ud\left(\cos\theta\right)} \end{eqnarray*} \end_inset \end_layout \begin_layout Standard The expressions \begin_inset Formula $\left(\sin\theta\right)^{-1}$ \end_inset and \begin_inset Formula $\frac{\ud P_{n}^{m}\left(\cos\theta\right)}{\ud\left(\cos\theta\right)}$ \end_inset are singular for \begin_inset Formula $\cos\theta=\pm1$ \end_inset , the limits \begin_inset Formula $\tau_{mn}\left(\pm1\right),\pi_{mn}\left(\pm1\right)$ \end_inset however exist. Labeling \begin_inset Formula $x\equiv\cos\theta$ \end_inset , \begin_inset Formula $\sqrt{\left(1+x\right)\left(1-x\right)}=\sqrt{1-x^{2}}\equiv\sin\theta$ \end_inset and using the asymptotic expression (DLMF 14.8.2) we obtain that the limits are nonzero only for \begin_inset Formula $m=\pm1$ \end_inset and \begin_inset Formula \begin{eqnarray*} \pi_{1\nu}(+1-) & = & CS\frac{\nu\left(\nu+1\right)}{2}\\ \tau_{1\nu}(+1-) & = & CS\frac{\nu\left(\nu+1\right)}{2} \end{eqnarray*} \end_inset and using the parity property \begin_inset Formula $P_{n}^{m}\left(-x\right)=\left(-1\right)^{m+n}P_{n}^{m}\left(x\right)$ \end_inset \begin_inset Formula \begin{eqnarray*} \pi_{1\nu}(-1+) & = & -CS\left(-1\right)^{\nu}\frac{\nu\left(\nu+1\right)}{2}\\ \tau_{1\nu}(-1+) & = & CS\left(-1\right)^{\nu}\frac{\nu\left(\nu+1\right)}{2} \end{eqnarray*} \end_inset For \begin_inset Formula $m=1$ \end_inset , we simply use the relation \begin_inset Formula $P_{n}^{-m}=\left(CS\right)^{m}P_{n}^{m}\frac{\left(n-m\right)!}{\left(n+m\right)!}$ \end_inset to get \begin_inset Formula \begin{eqnarray*} \pi_{-1\nu}(+1-) & = & \frac{CS}{2}\\ \tau_{-1\nu}(+1-) & = & -\frac{CS}{2}\\ \pi_{-1\nu}(-1+) & = & -\left(-1\right)^{\nu}\frac{CS}{2}\\ \tau_{-1\nu}(-1+) & = & -\left(-1\right)^{\nu}\frac{CS}{2} \end{eqnarray*} \end_inset where \begin_inset Formula $CS$ \end_inset is \begin_inset Formula $-1$ \end_inset if the Condon-Shortley phase is employed on the level of Legendre polynomials, 1 otherwise. \end_layout \begin_layout Subsection Taylor \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray*} \tilde{\pi}_{mn}\left(\cos\theta\right) & = & \sqrt{\frac{2n+1}{4\pi}\frac{\left(n-m\right)!}{\left(n+m\right)!}}\frac{m}{\sin\theta}P_{n}^{m}\left(\cos\theta\right)\\ \tilde{\tau}_{mn}\left(\cos\theta\right) & = & \sqrt{\frac{2n+1}{4\pi}\frac{\left(n-m\right)!}{\left(n+m\right)!}}\frac{\ud}{\ud\theta}P_{n}^{m}\left(\cos\theta\right) \end{eqnarray*} \end_inset \end_layout \begin_layout Standard The limiting expressions are obtained simply by multiplying the expressions from sec. \begin_inset CommandInset ref LatexCommand ref reference "sub:Xu pitau" \end_inset by the normalisation factor, \begin_inset Formula \begin{eqnarray*} \tilde{\pi}_{1\nu}(+1-) & = & CS\sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}}{2}\\ \tilde{\tau}_{1\nu}(+1-) & = & CS\sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}}{2}\\ \tilde{\pi}_{1\nu}(-1+) & = & -CS\left(-1\right)^{\nu}\sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}}{2}\\ \tilde{\tau}_{1\nu}(-1+) & = & CS\left(-1\right)^{\nu}\sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}}{2} \end{eqnarray*} \end_inset \begin_inset Formula \begin{eqnarray*} \tilde{\pi}_{-1\nu}(+1-) & = & CS\sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}}{2}\\ \tilde{\tau}_{-1\nu}(+1-) & = & -CS\sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}}{2}\\ \tilde{\pi}_{-1\nu}(-1+) & = & -CS\left(-1\right)^{\nu}\sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}\left(\nu+2\right)}{2}\\ \tilde{\tau}_{-1\nu}(-1+) & = & -CS\left(-1\right)^{\nu}\sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}\left(\nu+2\right)}{2} \end{eqnarray*} \end_inset i.e. the expressions for \begin_inset Formula $m=-1$ \end_inset are the same as for \begin_inset Formula $m=1$ \end_inset except for the sign if Condon-Shortley phase is used on the Legendre polynomial level. \end_layout \begin_layout Section Vector spherical harmonics (?) \end_layout \begin_layout Subsection Kristensson \end_layout \begin_layout Standard Original formulation, sect. [K]D.3.3 \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray} \vect A_{1lm}\left(\hat{\vect r}\right) & = & \frac{1}{\sqrt{l\left(l+1\right)}}\left(\hat{\vect{\theta}}\frac{1}{\sin\theta}\frac{\partial}{\partial\phi}Y_{lm}\left(\hat{\vect r}\right)-\hat{\vect{\phi}}\frac{\partial}{\partial\theta}Y_{lm}\left(\hat{\vect r}\right)\right)\nonumber \\ & = & \frac{1}{\sqrt{l\left(l+1\right)}}\nabla\times\left(\vect rY_{lm}\left(\hat{\vect r}\right)\right)\nonumber \\ \vect A_{2lm}\left(\hat{\vect r}\right) & = & \frac{1}{\sqrt{l\left(l+1\right)}}\left(\hat{\vect{\theta}}\frac{\partial}{\partial\phi}Y_{lm}\left(\hat{\vect r}\right)-\hat{\vect{\phi}}\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}Y_{lm}\left(\hat{\vect r}\right)\right)\label{eq:vector spherical harmonics Kristensson}\\ & = & \frac{1}{\sqrt{l\left(l+1\right)}}r\nabla Y_{lm}\left(\hat{\vect r}\right)\nonumber \\ \vect A_{3lm}\left(\hat{\vect r}\right) & = & \hat{\vect r}Y_{lm}\left(\hat{\vect r}\right)\nonumber \end{eqnarray} \end_inset Normalisation: \begin_inset Formula \[ \int\vect A_{n}\left(\hat{\vect r}\right)\cdot\vect A_{n'}^{\dagger}\left(\hat{\vect r}\right)\,\ud\Omega=\delta_{nn'} \] \end_inset Here \begin_inset Formula $\mbox{ }^{\dagger}$ \end_inset means just complex conjugate, apparently (see footnote on p. 89). \end_layout \begin_layout Subsection Jackson \end_layout \begin_layout Standard \begin_inset CommandInset citation LatexCommand cite after "(9.101)" key "jackson_classical_1998" \end_inset : \begin_inset Formula \[ \vect X_{lm}(\theta,\phi)=\frac{1}{\sqrt{l(l+1)}}\vect LY_{lm}(\theta,\phi) \] \end_inset where \begin_inset CommandInset citation LatexCommand cite after "(9.119)" key "jackson_classical_1998" \end_inset \begin_inset Formula \[ \vect L=\frac{1}{i}\left(\vect r\times\vect{\nabla}\right) \] \end_inset for its expression in spherical coordinates and other properties check Jackson's book around the definitions. \end_layout \begin_layout Standard Normalisation \begin_inset CommandInset citation LatexCommand cite after "(9.120)" key "jackson_classical_1998" \end_inset : \begin_inset Formula \[ \int\vect X_{l'm'}^{*}\cdot\vect X_{lm}\,\ud\Omega=\delta_{ll'}\delta_{mm'} \] \end_inset \end_layout \begin_layout Standard Local sum rule \begin_inset CommandInset citation LatexCommand cite after "(9.153)" key "jackson_classical_1998" \end_inset : \begin_inset Formula \[ \sum_{m=-l}^{l}\left|\vect X_{lm}(\theta,\phi)^{2}\right|=\frac{2l+1}{4\pi} \] \end_inset \end_layout \begin_layout Section Spherical Bessel functions \begin_inset CommandInset label LatexCommand label name "sec:Spherical-Bessel-functions" \end_inset \end_layout \begin_layout Standard Cf. [DLMF] §10.47–60. \end_layout \begin_layout Standard The radial dependence of spherical vector waves is given by the spherical Bessel functions and their first derivatives. Commonly, the following notation is adopted \begin_inset Formula \begin{eqnarray*} z_{n}^{(1)}(x) & = & j_{n}(x),\\ z_{n}^{(2)}(x) & = & y_{n}(x),\\ z_{n}^{(3)}(x) & = & h_{n}^{(1)}(x)=j_{n}(x)+iy_{n}(x),\\ z_{n}^{(4)}(x) & = & h_{n}^{(2)}(x)=j_{n}(x)-iy_{n}(x). \end{eqnarray*} \end_inset Here, \begin_inset Formula $j_{n}$ \end_inset is the spherical Bessel function of first kind (regular), \begin_inset Formula $y_{j}$ \end_inset is the spherical Bessel function of second kind (singular), and \begin_inset Formula $h_{n}^{(1)},h_{n}^{(2)}$ \end_inset are the Hankel functions a.k.a. spherical Bessel functions of third kind. In spherical vector waves, \begin_inset Formula $j_{n}$ \end_inset corresponds to regular waves, \begin_inset Formula $h^{(1)}$ \end_inset corresponds (by the usual convention) to outgoing waves, and \begin_inset Formula $h^{(2)}$ \end_inset corresponds to incoming waves. To describe scattering, we need two sets of waves with two different types of spherical Bessel functions \begin_inset Formula $z_{n}^{(J)}$ \end_inset . Most common choice is \begin_inset Formula $J=1,3$ \end_inset , because if we decompose the field into spherical waves centered at \begin_inset Formula $\vect r_{0}$ \end_inset , the field produced by other sources (e.g. spherical waves from other scatterers or a plane wave) is always regular at \begin_inset Formula $\vect r_{0}$ \end_inset . Second choice which makes a bit of sense is \begin_inset Formula $J=3,4$ \end_inset as it leads to a nice expression for the energy transport. \end_layout \begin_layout Subsection Limiting Forms \end_layout \begin_layout Standard [DLMF] §10.52: \end_layout \begin_layout Subsection \begin_inset Formula $z\to0$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray*} j_{n}(z) & \sim & z^{n}/(2n+1)!!\\ h_{n}^{(1)}(z)\sim iy(z) & \sim & -i\left(2n+1\right)!!/z^{n+1} \end{eqnarray*} \end_inset \end_layout \begin_layout Section Spherical vector waves \end_layout \begin_layout Standard TODO \begin_inset Formula $M,N,\psi,\chi,\widetilde{M},\widetilde{N},u,v,w,\dots$ \end_inset , sine/cosine convention (B&H), ... \end_layout \begin_layout Standard There are two mutually orthogonal types of divergence-free (everywhere except in the origin for singular waves) spherical vector waves, which I call electric and magnetic, given by the type of multipole source to which they correspond. This is another distinction than the regular/singular/ingoing/outgoing waves given by the type of the radial dependence (cf. section \begin_inset CommandInset ref LatexCommand ref reference "sec:Spherical-Bessel-functions" \end_inset ). Oscillating electric current in a tiny rod parallel to its axis will generate electric dipole waves (net dipole moment of magnetic current is zero) moment , whereas oscillating electric current in a tiny circular loop will generate magnetic dipole waves (net dipole moment of electric current is zero). \end_layout \begin_layout Standard In the usual cases we encounter, the part described by the magnetic waves is pretty small. \end_layout \begin_layout Subsection Taylor \end_layout \begin_layout Standard Definition [T](2.40); \begin_inset Formula $\widetilde{\vect N}_{mn}^{(j)},\widetilde{\vect M}_{mn}^{(j)}$ \end_inset are the electric and magnetic waves, respectively: \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray*} \widetilde{\vect N}_{mn}^{(j)} & = & \frac{n(n+1)}{kr}\sqrt{\frac{2n+1}{4\pi}\frac{\left(n-m\right)!}{\left(n+m\right)!}}P_{n}^{m}\left(\cos\theta\right)e^{im\phi}z_{n}^{j}\left(kr\right)\hat{\vect r}\\ & & +\left[\tilde{\tau}_{mn}\left(\cos\theta\right)\hat{\vect{\theta}}+i\tilde{\pi}_{mn}\left(\cos\theta\right)\hat{\vect{\phi}}\right]e^{im\phi}\frac{1}{kr}\frac{\ud\left(kr\,z_{n}^{j}\left(kr\right)\right)}{\ud(kr)}\\ \widetilde{\vect M}_{mn}^{(j)} & = & \left[i\tilde{\pi}_{mn}\left(\cos\theta\right)\hat{\vect{\theta}}-\tilde{\tau}_{mn}\left(\cos\theta\right)\hat{\vect{\phi}}\right]e^{im\phi}z_{n}^{j}\left(kr\right) \end{eqnarray*} \end_inset \end_layout \begin_layout Subsection Xu \end_layout \begin_layout Standard are the electric and magnetic waves, respectively: \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray*} \vect N_{mn}^{(j)} & = & \frac{n(n+1)}{kr}P_{n}^{m}\left(\cos\theta\right)e^{im\phi}z_{n}^{j}\left(kr\right)\hat{\vect r}\\ & & +\left[\tau_{mn}\left(\cos\theta\right)\hat{\vect{\theta}}+i\pi_{mn}\left(\cos\theta\right)\hat{\vect{\phi}}\right]e^{im\phi}\frac{1}{kr}\frac{\ud\left(kr\,z_{n}^{j}\left(kr\right)\right)}{\ud(kr)}\\ \vect M_{mn}^{(j)} & = & \left[i\pi_{mn}\left(\cos\theta\right)\hat{\vect{\theta}}-\tau_{mn}\left(\cos\theta\right)\hat{\vect{\phi}}\right]e^{im\phi}z_{n}^{j}\left(kr\right) \end{eqnarray*} \end_inset \end_layout \begin_layout Subsection Kristensson \end_layout \begin_layout Standard Definition [K](2.4.6); \begin_inset Formula $\vect u_{\tau lm},\vect v_{\tau lm},\vect w_{\tau lm}$ \end_inset are the waves with \begin_inset Formula $j=3,1,4$ \end_inset respectively, i.e. outgoing, regular and incoming waves. The first index distinguishes between the electric ( \begin_inset Formula $\tau=2$ \end_inset ) and magnetic ( \begin_inset Formula $\tau=1$ \end_inset ). Kristensson uses a multiindex \begin_inset Formula $n\equiv(\tau,l,m)$ \end_inset to simlify the notation. \begin_inset Formula \begin{eqnarray*} \left(\vect{u/v/w}\right)_{2lm} & = & \frac{1}{kr}\frac{\ud\left(kr\,z_{l}^{(j)}\left(kr\right)\right)}{\ud\,kr}\vect A_{2lm}\left(\hat{\vect r}\right)+\sqrt{l\left(l+1\right)}\frac{z_{l}^{(j)}(kr)}{kr}\vect A_{3lm}\left(\hat{\vect r}\right)\\ \left(\vect{u/v/w}\right)_{1lm} & = & z_{l}^{(j)}\left(kr\right)\vect A_{1lm}\left(\hat{\vect r}\right) \end{eqnarray*} \end_inset \end_layout \begin_layout Subsection Kristensson vs. Xu \end_layout \begin_layout Standard As in \begin_inset CommandInset citation LatexCommand cite after "eq. (36)" key "xu_calculation_1996" \end_inset with unnormalised Legendre polynomials: \begin_inset Formula \begin{eqnarray*} \left(\vect{u/v/w}\right)_{1lm} & = & \left(\mbox{CS}\right)^{m}\sqrt{\frac{2l+1}{4\pi}\frac{\left(l-m\right)!}{\left(l+m\right)!}}\frac{\vect N_{ml}^{(3/1/4)}}{\sqrt{l\left(l+1\right)}}\\ \left(\vect{u/v/w}\right)_{1lm} & = & \left(\mbox{CS}\right)^{m}\sqrt{\frac{2l+1}{4\pi}\frac{\left(l-m\right)!}{\left(l+m\right)!}}\frac{\vect M_{ml}^{(3/1/4)}}{\sqrt{l\left(l+1\right)}} \end{eqnarray*} \end_inset where CS is \begin_inset Formula $-1$ \end_inset in Kristensson's text. N.B. be careful about the translation coefficients and \begin_inset CommandInset citation LatexCommand cite after "eq. (81)" key "xu_calculation_1996" \end_inset , Xu's text is a bit confusing. \end_layout \begin_layout Subsection Relation between Kristensson and Taylor \begin_inset CommandInset label LatexCommand label name "sub:Kristensson-v-Taylor" \end_inset \end_layout \begin_layout Standard Kristensson's and Taylor's VSWFs seem to differ only by an \begin_inset Formula $l$ \end_inset -dependent normalization factor, and notation of course (n.b. the inverse index order) \begin_inset Formula \begin{eqnarray*} \left(\vect{u/v/w}\right)_{2lm} & = & \frac{\widetilde{\vect N}_{ml}^{(3/1/4)}}{\sqrt{l\left(l+1\right)}}\\ \left(\vect{u/v/w}\right)_{1lm} & = & \frac{\widetilde{\vect M}_{ml}^{(3/1/4)}}{\sqrt{l\left(l+1\right)}} \end{eqnarray*} \end_inset \end_layout \begin_layout Section Plane wave expansion \end_layout \begin_layout Subsection Taylor \end_layout \begin_layout Standard \begin_inset Formula $x$ \end_inset -polarised, \begin_inset Formula $z$ \end_inset -propagating plane wave, \begin_inset Formula $\vect E=E_{0}\hat{\vect x}e^{i\vect k\cdot\hat{\vect z}}$ \end_inset (CHECK): \begin_inset Formula \begin{eqnarray*} \vect E & = & -i\left(p_{mn}\widetilde{\vect N}_{mn}^{(1)}+q_{mn}\widetilde{\vect M}_{mn}^{(1)}\right)\\ p_{mn} & = & E_{0}\frac{4\pi i^{n}}{n(n+1)}\tilde{\tau}_{mn}(1)\\ q_{mn} & = & E_{0}\frac{4\pi i^{n}}{n(n+1)}\tilde{\pi}_{mn}(1) \end{eqnarray*} \end_inset while it can be shown that \begin_inset Formula \begin{eqnarray*} \tilde{\pi}_{mn}(1) & = & -\frac{1}{2}\sqrt{\frac{\left(2n+1\right)\left(n\left(n+1\right)\right)}{4\pi}}\left(\delta_{m,1}+\delta_{m,-1}\right)\\ \tilde{\tau}_{mn}(1) & = & -\frac{1}{2}\sqrt{\frac{\left(2n+1\right)\left(n\left(n+1\right)\right)}{4\pi}}\left(\delta_{m,1}-\delta_{m,-1}\right) \end{eqnarray*} \end_inset \end_layout \begin_layout Subsection Kristensson \end_layout \begin_layout Standard \begin_inset Formula $x$ \end_inset -polarised, \begin_inset Formula $z$ \end_inset -propagating plane wave, \begin_inset Formula $\vect E=E_{0}\hat{\vect x}e^{i\vect k\cdot\hat{\vect z}}$ \end_inset (CHECK, ): \begin_inset Formula \[ \vect E=\sum_{n}a_{n}\vect v_{n} \] \end_inset \begin_inset Formula \begin{eqnarray*} a_{1lm} & = & E_{0}i^{l+1}\sqrt{\left(2l+1\right)\pi}\left(\delta_{m,1}+\delta_{m,-1}\right)\\ a_{2lm} & = & E_{0}i^{l+1}\sqrt{\left(2l+1\right)\pi}\left(\delta_{m,1}+\delta_{m,-1}\right) \end{eqnarray*} \end_inset \end_layout \begin_layout Section Radiated energy \end_layout \begin_layout Standard In this section I summarize the formulae for power \begin_inset Formula $P$ \end_inset radiated from the system. For an absorbing scatterer, this should be negative (n.b. sign conventions can be sometimes confusing). If the system is excited by a plane wave with intensity \begin_inset Formula $E_{0}$ \end_inset , this can be used to calculate the absorption cross section (TODO check if it should be multiplied by the 2), \begin_inset Formula \[ \sigma_{\mathrm{abs}}=-\frac{2P}{\varepsilon\varepsilon_{0}\left|E_{0}\right|^{2}}. \] \end_inset \end_layout \begin_layout Subsection Kristensson \begin_inset CommandInset label LatexCommand label name "sub:Radiated enenergy-Kristensson" \end_inset \end_layout \begin_layout Standard Sect. [K]2.6.2; here this form of expansion is assumed: \begin_inset Formula \begin{equation} \vect E\left(\vect r,\omega\right)=k\sqrt{\eta_{0}\eta}\sum_{n}\left(a_{n}\vect v_{n}\left(k\vect r\right)+f_{n}\vect u_{n}\left(k\vect r\right)\right).\label{eq:power-Kristensson-E} \end{equation} \end_inset Here \begin_inset Formula $\eta_{0}=\sqrt{\mu_{0}/\varepsilon_{0}}$ \end_inset is the wave impedance of free space and \begin_inset Formula $\eta=\sqrt{\mu/\varepsilon}$ \end_inset is the relative wave impedance of the medium. \end_layout \begin_layout Standard The radiated power is then (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 The first term is obviously the power radiated away by the outgoing waves. The second term must then be minus the power sucked by the scatterer from the exciting wave. If the exciting wave is plane, it gives us the extinction cross section \begin_inset Formula \[ \sigma_{\mathrm{tot}}=-\frac{\sum_{n}\Re\left(f_{n}a_{n}^{*}\right)}{\varepsilon\varepsilon_{0}\left|E_{0}\right|^{2}} \] \end_inset \end_layout \begin_layout Subsection Taylor \end_layout \begin_layout Standard Here I derive the radiated power in Taylor's convention by applying the relations from subsection \begin_inset CommandInset ref LatexCommand ref reference "sub:Kristensson-v-Taylor" \end_inset to the Kristensson's formulae (sect. \begin_inset CommandInset ref LatexCommand ref reference "sub:Radiated enenergy-Kristensson" \end_inset ). \end_layout \begin_layout Standard Assume the external field decomposed as (here I use tildes even for the expansion coefficients in order to avoid confusion with the \begin_inset Formula $a_{n}$ \end_inset in \begin_inset CommandInset ref LatexCommand eqref reference "eq:power-Kristensson-E" \end_inset ) \begin_inset Formula \[ \vect E\left(\vect r,\omega\right)=\sum_{mn}\left[-i\left(\tilde{p}_{mn}\vect{\widetilde{N}}_{mn}^{(1)}+\tilde{q}_{mn}\widetilde{\vect M}_{mn}^{(1)}\right)+i\left(\tilde{a}_{mn}\widetilde{\vect N}_{mn}^{(3)}+\tilde{b}_{mn}\widetilde{\vect M}_{mn}^{(3)}\right)\right] \] \end_inset (there is minus between the regular and outgoing part!). The coefficients are related to those from \begin_inset CommandInset ref LatexCommand eqref reference "eq:power-Kristensson-E" \end_inset as \begin_inset Formula \[ \tilde{p}_{mn}=\frac{k\sqrt{\eta_{0}\eta}}{-i\sqrt{n(n+1)}}a_{2nm},\quad\tilde{q}_{mn}=\frac{k\sqrt{\eta_{0}\eta}}{-i\sqrt{n(n+1)}}a_{1nm}, \] \end_inset \begin_inset Formula \[ \tilde{a}_{mn}=\frac{k\sqrt{\eta_{0}\eta}}{i\sqrt{n(n+1)}}f_{2nm},\quad\tilde{b}_{mn}=\frac{k\sqrt{\eta_{0}\eta}}{i\sqrt{n(n+1)}}f_{1nm}. \] \end_inset The radiated power is then \begin_inset Formula \[ P=\frac{1}{2}\sum_{m,n}\frac{n\left(n+1\right)}{k^{2}\eta_{0}\eta}\left(\left|a_{mn}\right|^{2}+\left|b_{mn}\right|^{2}-\Re\left(a_{mn}p_{mn}^{*}\right)-\Re\left(b_{mn}q_{mn}^{*}\right)\right). \] \end_inset If the exciting wave is a plane wave, the extinction cross section is \begin_inset Formula \[ \sigma_{\mathrm{tot}}=\frac{1}{\varepsilon\varepsilon_{0}\left|E_{0}\right|^{2}k^{2}\eta_{0}\eta}\sum_{m,n}n(n+1)\left(\Re\left(a_{mn}p_{mn}^{*}\right)+\Re\left(b_{mn}q_{mn}^{*}\right)\right) \] \end_inset \end_layout \begin_layout Subsection Jackson \end_layout \begin_layout Standard \begin_inset CommandInset citation LatexCommand cite after "(9.155)" key "jackson_classical_1998" \end_inset : \begin_inset Formula \[ P=\frac{Z_{0}}{2k^{2}}\sum_{l,m}\left[\left|a_{E}(l,m)\right|^{2}+\left|a_{M}(l,m)\right|^{2}\right] \] \end_inset \end_layout \begin_layout Section Limit solutions \end_layout \begin_layout Subsection Far-field asymptotic solution \end_layout \begin_layout Standard TODO start from \begin_inset CommandInset citation LatexCommand cite after "(A7)" key "pustovit_plasmon-mediated_2010" \end_inset and Jackson (9.169) and around. \end_layout \begin_layout Subsection Near field limit \end_layout \begin_layout Chapter Single particle scattering and Mie theory \end_layout \begin_layout Standard The basic idea is simple. For an exciting spherical wave (usually the regular wave in whatever convention ) of a given frequency \begin_inset Formula $\omega$ \end_inset , type \begin_inset Formula $\zeta'$ \end_inset (electric or magnetic), degree \begin_inset Formula $l'$ \end_inset and order \begin_inset Formula $m'$ \end_inset , the particle responds with waves from the complementary set (e.g. outgoing waves), with the same frequency \begin_inset Formula $\omega$ \end_inset , but any type \begin_inset Formula $\zeta$ \end_inset , degree \begin_inset Formula $l$ \end_inset and order \begin_inset Formula $m$ \end_inset , in a way that the Maxwell's equations are satisfied, with the coefficients \begin_inset Formula $T_{l,m;l',m'}^{\zeta,\zeta'}(\omega)$ \end_inset . This yields one row in the scattering matrix (often called the \begin_inset Formula $T$ \end_inset -matrix) \begin_inset Formula $T(\omega)$ \end_inset , which fully characterizes the scattering properties of the particle (in the linear regime, of course). Analytical expression for the matrix is known for spherical scatterer, otherwise it is computed numerically (using DDA, BEM or whatever). So if we have the two sets of spherical wave functions \begin_inset Formula $\vect f_{lm}^{J_{1},\zeta}$ \end_inset , \begin_inset Formula $\vect f_{lm}^{J_{2},\zeta}$ \end_inset and the full \begin_inset Quotes sld \end_inset exciting \begin_inset Quotes srd \end_inset wave has electric field given as \begin_inset Formula \[ \vect E_{\mathrm{inc}}=\sum_{\zeta'=\mathrm{E,M}}\sum_{l',m'}c_{l'm'}^{\zeta'}\vect f_{l'm'}^{\zeta'}, \] \end_inset the \begin_inset Quotes sld \end_inset scattered \begin_inset Quotes srd \end_inset field will be \begin_inset Formula \[ \vect E_{\mathrm{scat}}=\sum_{\zeta',l',m'}\sum_{\zeta,l,m}T_{l,m;l',m'}^{\zeta,\zeta'}c_{l'm'}^{\zeta'}\vect f_{lm}^{\zeta}, \] \end_inset and the total field around the scaterer is \begin_inset Formula $\vect E=\vect E_{\mathrm{ext}}+\vect E_{\mathrm{scat}}$ \end_inset . \end_layout \begin_layout Section Mie theory – full version \end_layout \begin_layout Standard \begin_inset Formula $T$ \end_inset -matrix for a spherical particle is type-, degree- and order- diagonal, that is, \begin_inset Formula $T_{l',m';l,m}^{\zeta',\zeta}(\omega)=0$ \end_inset if \begin_inset Formula $l\ne l'$ \end_inset , \begin_inset Formula $m\ne m'$ \end_inset or \begin_inset Formula $\zeta\ne\zeta'$ \end_inset . Moreover, it does not depend on \begin_inset Formula $m$ \end_inset , so \begin_inset Formula \[ T_{l,m;l',m'}^{\zeta,\zeta'}(\omega)=T_{l}^{\zeta}\left(\omega\right)\delta_{\zeta'\zeta}\delta_{l'l}\delta_{m'm} \] \end_inset where for the usual choice \begin_inset Formula $J_{1}=1,J_{2}=3$ \end_inset \begin_inset Formula \begin{eqnarray*} T_{l}^{E}\left(\omega\right) & = & TODO,\\ T_{l}^{M}(\omega) & = & TODO. \end{eqnarray*} \end_inset \end_layout \begin_layout Section Long wave approximation for spherical nanoparticle \end_layout \begin_layout Standard TODO start from \begin_inset CommandInset citation LatexCommand cite after "(A11)" key "pustovit_plasmon-mediated_2010" \end_inset and around. \end_layout \begin_layout Section Note on transforming T-matrix conventions \end_layout \begin_layout Standard T-matrix depends on the used conventions as well. This is not apparent for the Mie case as the T-matrix for a sphere is \begin_inset Quotes sld \end_inset diagonal \begin_inset Quotes srd \end_inset . But for other shapes, dipole incoming field can induce also higher-order multipoles in the nanoparticle, etc. The easiest way to determine the transformation properties is to write down the total scattered electric field for both conventions in the form \begin_inset Formula \[ \vect E_{\mathrm{scat}}=\sum_{n'}\sum_{n}T_{n'}^{n}c^{n'}\vect f_{n}=\sum_{n'}\sum_{n}\widetilde{T}_{n'}^{n}\widetilde{c}^{n'}\widetilde{\vect f}_{n} \] \end_inset where we merged all the indices into single multiindex \begin_inset Formula $n$ \end_inset or \begin_inset Formula $n'$ \end_inset . This way of writing immediately suggest how to transform the T-matrix into the new convention if we know the transformation properties of the base waves and expansion coefficients, as it reminds the notation used in geometry – \begin_inset Formula $c^{\alpha}$ \end_inset are \begin_inset Quotes sld \end_inset vector coordinates \begin_inset Quotes srd \end_inset , \begin_inset Formula $\vect f_{\alpha}$ \end_inset are \begin_inset Quotes sld \end_inset base vectors \begin_inset Quotes srd \end_inset . Obviously, T-matrix is then \begin_inset Quotes sld \end_inset tensor of type (1,1) \begin_inset Quotes srd \end_inset , and it transforms as vector coordinates (i.e. wave expansion coefficients) in the \begin_inset Formula $n$ \end_inset (outgoing wave) indices and as form coordinates in the \begin_inset Formula $n'$ \end_inset (regular/illuminating wave) indices. Form coordinates change in the same waves as base vectors \end_layout \begin_layout Subsection Kristensson to Taylor \end_layout \begin_layout Standard For instance, let us transform between from the Kristensson's to Taylor's convention. We know that the Taylor's base vectors are \begin_inset Quotes sld \end_inset larger \begin_inset Quotes srd \end_inset : \begin_inset Formula $\widetilde{\vect N}_{ml}^{(3/1/4)}=\sqrt{l(l+1)}\left(\vect{u/v/w}\right)_{2lm}$ \end_inset etc, so the coefficients must be smaller by the reciprocal factor, e.g. \begin_inset Formula $\tilde{a}_{ml}=f_{2lm}/\sqrt{l(l+1)}$ \end_inset (now we assume that there are no other prefactors in the expansion of the field, they are already included in the coefficients). Then the T-matrix in the Taylor's convention (tilded) can be calculated from the T-matrix in the Kristensson's convention as \begin_inset Formula \[ \underbrace{\widetilde{T}_{\zeta'l'm'}^{\zeta lm}}_{\mbox{Taylor}}=\frac{\sqrt{l'(l'+1)}}{\sqrt{l(l+1)}}\underbrace{T_{\zeta'l'm'}^{\zeta lm}}_{\mbox{Krist.}}\,_{\leftarrow\mbox{illuminating}}^{\leftarrow\mbox{outgoing}}. \] \end_inset \end_layout \begin_layout Subsubsection scuff-tmatrix output \end_layout \begin_layout Standard Indices of the outgoing wave (without primes) come first, illuminating regular wave (with primes) second in the output files of scuff-tmatrix. It seems that it at least in the electric part, the output of scuff-tmatrix is equivalent to the Kristensson's convention. Not sure whether it is also true for the E-M cross terms. \end_layout \begin_layout Chapter Green's functions \end_layout \begin_layout Section xyz pure free-space dipole waves in terms of SVWF \end_layout \begin_layout Section Mie decomposition of Green's function for single nanoparticle \end_layout \begin_layout Chapter Translation of spherical waves: getting insane \end_layout \begin_layout Chapter Multiple scattering: nice linear algebra born from all the mess \end_layout \begin_layout Chapter Quantisation of quasistatic modes of a sphere \end_layout \begin_layout Standard \begin_inset CommandInset bibtex LatexCommand bibtex bibfiles "Electrodynamics,/home/mmn/repo/qpms/Electrodynamics" options "plain" \end_inset \end_layout \end_body \end_document