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For more info see http://www.lyx.org/ \lyxformat 413 \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_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_amsmath 1 \use_esint 1 \use_mhchem 1 \use_mathdots 1 \cite_engine basic \use_bibtopic false \use_indices false \paperorientation portrait \suppress_date false \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 Subsection Kristensson \end_layout \begin_layout Standard \begin_inset Formula \[ P_{l}^{-m}=\left(-1\right)^{m}\frac{\left(l-m\right)!}{\left(l+m\right)!}P_{l}^{m}\left(\cos\theta\right),\quad m\ge0 \] \end_inset 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 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 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)\\ \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)\\ \vect A_{3lm}\left(\hat{\vect r}\right) & = & \hat{\vect r}Y_{lm}\left(\hat{\vect r}\right) \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 Section Spherical Bessel functions \begin_inset CommandInset label LatexCommand label name "sec:Spherical-Bessel-functions" \end_inset \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 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}z_{n}^{j}\left(kr\right)\\ \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 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 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, \begin_inset Formula \[ \sigma_{\mathrm{abs}}=-\frac{P}{\varepsilon\varepsilon_{0}\left|E_{0}\right|^{2}}. \] \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 \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 \[ P=\frac{1}{2}\sum_{m,n}\frac{n\left(n+1\right)}{k^{2}\eta_{0}}\left(\left|a_{mn}\right|^{2}+\left|b_{mn}\right|^{2}-\Re\left(a_{mn}p_{mn}^{*}\right)-\Re\left(b_{mn}q_{mn}^{*}\right)\right). \] \end_inset \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 \end_layout \begin_layout Subsection Near field limit \end_layout \begin_layout Chapter Mie Theory \end_layout \begin_layout Section Full version \end_layout \begin_layout Section Long wave approximation \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 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 Standard \begin_inset CommandInset bibtex LatexCommand bibtex bibfiles "dipdip" options "plain" \end_inset \end_layout \end_body \end_document