From fe0ccb92b8ab7f8872ed5af3672d0ad642f831a5 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Thu, 30 Jun 2016 16:02:31 +0300 Subject: [PATCH] Adding some formulae to the text Former-commit-id: c244a6681365147a17017dbb1f0f75192d99c351 --- Scattering and Shit.lyx | 554 +++++++++++++++++++++++++++++++++++++++- 1 file changed, 548 insertions(+), 6 deletions(-) diff --git a/Scattering and Shit.lyx b/Scattering and Shit.lyx index 2e39e81..ec24821 100644 --- a/Scattering and Shit.lyx +++ b/Scattering and Shit.lyx @@ -79,8 +79,21 @@ \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 shit +Electromagnetic multiple scattering, spherical waves and **** \end_layout \begin_layout Author @@ -95,33 +108,562 @@ Zillion conventions for spherical vector waves 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 (?), spherical Bessel functions +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 Section -All the conventions +\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 @@ -183,11 +725,11 @@ Mie decomposition of Green's function for single nanoparticle \end_layout \begin_layout Chapter -Translation of spherical waves: shit gets insane +Translation of spherical waves: getting insane \end_layout \begin_layout Chapter -Multiple scattering: nice linear algebra born from all the shit +Multiple scattering: nice linear algebra born from all the mess \end_layout \begin_layout Standard