#LyX 2.0 created this file. For more info see http://www.lyx.org/ \lyxformat 413 \begin_document \begin_header \textclass revtex4 \options pra,superscriptaddress,twocolumn,notitlepage \use_default_options false \maintain_unincluded_children false \language english \language_package default \inputencoding auto \fontencoding global \font_roman default \font_sans default \font_typewriter default \font_default_family default \use_non_tex_fonts true \font_sc false \font_osf false \font_sf_scale 100 \font_tt_scale 100 \graphics default \default_output_format pdf4 \output_sync 0 \bibtex_command bibtex \index_command default \paperfontsize default \spacing single \use_hyperref false \papersize a4paper \use_geometry false \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 \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 \lang finnish \begin_inset FormulaMacro \newcommand{\ket}[1]{\left|#1\right\rangle } \end_inset \begin_inset FormulaMacro \newcommand{\bra}[1]{\left\langle #1\right|} \end_inset \lang english \begin_inset FormulaMacro \newcommand{\vect}[1]{\mathbf{\boldsymbol{#1}}} {\boldsymbol{\mathbf{#1}}} \end_inset \begin_inset FormulaMacro \newcommand{\uvec}[1]{\mathbf{\boldsymbol{\hat{#1}}}} {\boldsymbol{\hat{\mathbf{#1}}}} \end_inset \begin_inset FormulaMacro \newcommand{\ud}{\mathrm{d}} \end_inset \end_layout \begin_layout Title Technical notes on quantum electromagnetic multiple scattering \end_layout \begin_layout Author Marek Nečada \end_layout \begin_layout Affiliation COMP Centre of Excellence, Department of Applied Physics, Aalto University, P.O. Box 15100, Fi-00076 Aalto, Finland \end_layout \begin_layout Date \begin_inset ERT status open \begin_layout Plain Layout \backslash today \end_layout \end_inset \end_layout \begin_layout Abstract ... \end_layout \begin_layout Section Theory of quantum electromagnetic multiple scattering \end_layout \begin_layout Subsection Incoherent pumping \end_layout \begin_layout Standard Cf. Wubs \begin_inset CommandInset citation LatexCommand cite key "wubs_multiple-scattering_2004" \end_inset , Delga \begin_inset CommandInset citation LatexCommand cite key "delga_quantum_2014,delga_theory_2014" \end_inset . \end_layout \begin_layout Subsection General initial states \end_layout \begin_layout Standard Look at \begin_inset CommandInset citation LatexCommand cite key "landau_computational_2015" \end_inset for an inspiration for solving the LS equation with an arbitrary initial state. \end_layout \begin_layout Section Computing classical Green's functions \end_layout \begin_layout Standard The formulae below might differ depending on the conventions used by various authors. For instance, Taylor \begin_inset CommandInset citation LatexCommand cite key "taylor_optical_2011" \end_inset uses normalized spherical wavefunctions \begin_inset Formula $\widetilde{\vect M}_{mn}^{(j)},\widetilde{\vect N}_{mn}^{(j)}$ \end_inset which are designed in a way that avoids float number overflow of some of the variables during the numerical calculation. \end_layout \begin_layout Standard Beware of various conventions in definitions of Legendre functions etc. (the implementation in py-gmm differs, for example, by a factor of \begin_inset Formula $(-1)^{m}$ \end_inset from scipy.special.lpmn. I think this is also the reason that lead to the \begin_inset Quotes eld \end_inset wrong \begin_inset Quotes erd \end_inset signs in the addition coefficients in my code compared to \begin_inset CommandInset citation LatexCommand cite key "xu_calculation_1996" \end_inset . \end_layout \begin_layout Subsection T-Matrix method \end_layout \begin_layout Subsubsection VSWF decomposition \end_layout \begin_layout Standard Expressions for VSWF in Xu \begin_inset CommandInset citation LatexCommand cite after "(2)" key "xu_electromagnetic_1995" \end_inset : \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray} \vect M_{mn}^{(J)} & = & \left(i\uvec{\theta}\pi_{mn}(\cos\theta)-\uvec{\phi}\tau_{mn}(\cos\theta)\right)z_{n}^{(J)}(kr)e^{im\phi},\nonumber \\ \vect N_{mn}^{(J)} & = & \uvec rn(n+1)P_{n}^{m}(\cos\theta)\frac{z_{n}^{(J)}(kr)}{kr}e^{im\phi}\label{eq:vswf}\\ & & +\left(\uvec{\theta}\tau_{mn}(\cos\theta)+i\uvec{\phi}\pi_{mn}(\cos\theta)\right)\nonumber \\ & & \phantom{+}\times\frac{1}{kr}\frac{\ud\left(rz_{n}^{(J)}(kr)\right)}{\ud r}e^{im\phi},\nonumber \\ & = & ...\nonumber \end{eqnarray} \end_inset where \begin_inset Formula $z_{n}^{(J)}$ \end_inset denotes \begin_inset Formula $j_{n},y_{n},h_{n}^{+},h_{n}^{-}$ \end_inset for \begin_inset Formula $J=1,2,3,4$ \end_inset , respectively, and \begin_inset Formula \begin{eqnarray*} \pi_{mn}(\cos\theta) & = & \frac{m}{\sin\theta}P_{n}^{m}(\cos\theta),\\ \tau_{mn}(\cos\theta) & = & \frac{\ud P_{n}^{m}(\cos\theta)}{\ud\theta}=-\sin\theta\frac{\ud P_{n}^{m}(\cos\theta)}{\ud\cos\theta}. \end{eqnarray*} \end_inset The expressions for \begin_inset Formula $\vect M_{mn}^{(J)},\vect N_{mn}^{(J)}$ \end_inset are dimensionless. \end_layout \begin_layout Standard \emph on Note about the case \begin_inset Formula $\theta\to0,\pi$ \end_inset : \emph default There is a divergent \begin_inset Formula $1/\sin\theta$ \end_inset factor in the \begin_inset Formula $\pi_{mn}(\cos\theta)$ \end_inset function. For \begin_inset Formula $m=0$ \end_inset , it is irrelevant because of the \begin_inset Formula $m$ \end_inset factor (it would be bad otherwise, because \begin_inset Formula $P_{n}^{0}(\cos\theta)$ \end_inset does not go to zero at \begin_inset Formula $\theta\to0,\pi$ \end_inset ). For \begin_inset Formula $\left|m\right|\ge2$ \end_inset , \begin_inset Formula $P_{n}^{m}(x)$ \end_inset behaves as \begin_inset Formula $o(x+1),o(x-1)$ \end_inset at \begin_inset Formula $-1,1$ \end_inset , so \begin_inset Formula $P_{n}^{m}(\cos\theta)$ \end_inset goes like \begin_inset Formula $o(\theta^{2}),o\left((\theta-\pi)^{2}\right)$ \end_inset at \begin_inset Formula $0,\pi$ \end_inset , which safely eliminates the divergent factor. However, for \begin_inset Formula $\left|m\right|=1$ \end_inset , the whole expression \begin_inset Formula $P_{n}^{m}(\cos\theta)/\sin\theta$ \end_inset has a finite nonzero limit for \begin_inset Formula $\theta\to0,\pi$ \end_inset . According to Mathematica (for \begin_inset Formula $\theta\to\pi,$ \end_inset Mathematica does not work well, but it can be derived from the \begin_inset Formula $\theta\to0$ \end_inset case and oddness/evenness). \begin_inset Formula \begin{eqnarray*} \lim_{\theta\to0}\frac{P_{n}^{1}(\cos\theta)}{\sin\theta} & = & -\frac{1}{2}n(1+n),\qquad\lim_{\theta\to0}\frac{P_{n}^{-1}(\cos\theta)}{\sin\theta}=\frac{1}{2},\\ \lim_{\theta\to\pi}\frac{P_{n}^{1}(\cos\theta)}{\sin\theta} & = & \frac{\left(-1\right)^{n}}{2}n(1+n),\qquad\lim_{\theta\to\pi}\frac{P_{n}^{-1}(\cos\theta)}{\sin\theta}=\frac{\left(-1\right)^{n+1}}{2}. \end{eqnarray*} \end_inset NOT COMPLETELY SURE ABOUT THE SIGN/NORMALIZATION CONVENTION HERE. IT HAS TO BE CHECKED. \end_layout \begin_layout Standard Expansions for the scattered fields are \begin_inset CommandInset citation LatexCommand cite after "(4)" key "xu_electromagnetic_1995" \end_inset : \begin_inset Formula \begin{eqnarray*} \vect E_{s}(j) & = & \sum_{n=1}^{\infty}\sum_{m=-n}^{n}iE_{mn}\left[a_{mn}^{j}\vect N_{mn}^{(3)}+b_{mn}^{j}\vect M_{mn}^{(3)}\right],\\ \vect H_{s}(j) & = & \frac{k}{\omega\mu}\sum_{n=1}^{\infty}\sum_{m=-n}^{n}E_{mn}\left[b_{mn}^{j}\vect N_{mn}^{(3)}+a_{mn}^{j}\vect M_{mn}^{(3)}\right]. \end{eqnarray*} \end_inset These expansions should be OK in SI units (take the Fourier transform of \begin_inset Formula $\nabla\times\vect E=-\partial\vect B/\partial t$ \end_inset and \begin_inset Formula $\vect B=\mu\vect H$ \end_inset ). For internal field of a sphere, the (regular-wave) expansion reads \begin_inset Formula \begin{eqnarray*} \vect E_{I}(j) & = & -\sum_{n=1}^{\infty}\sum_{m=-n}^{n}iE_{mn}\left[d_{mn}^{j}\vect N_{mn}^{(1)}+c_{mn}^{j}\vect M_{mn}^{(1)}\right],\\ \vect H_{I}(j) & = & -\frac{k}{\omega\mu}\sum_{n=1}^{\infty}\sum_{m=-n}^{n}E_{mn}\left[c_{mn}^{j}\vect N_{mn}^{(1)}+d_{mn}^{j}\vect M_{mn}^{(1)}\right] \end{eqnarray*} \end_inset (note the minus sign; I am not sure about its role) and the incident field (incl. field from the other scatterers) is assumed to have the same regular-wave form \begin_inset Formula \begin{eqnarray*} \vect E_{i}(j) & = & -\sum_{n=1}^{\infty}\sum_{m=-n}^{n}iE_{mn}\left[p_{mn}^{j}\vect N_{mn}^{(1)}+q_{mn}^{j}\vect M_{mn}^{(1)}\right],\\ \vect H_{i}(j) & = & -\frac{k}{\omega\mu}\sum_{n=1}^{\infty}\sum_{m=-n}^{n}E_{mn}\left[q_{mn}^{j}\vect N_{mn}^{(1)}+p_{mn}^{j}\vect M_{mn}^{(1)}\right]. \end{eqnarray*} \end_inset Note that \begin_inset Formula $k/\omega\mu=\sqrt{\varepsilon_{r}\varepsilon_{0}/\mu_{r}\mu_{0}}=1/\eta_{r}\eta_{0}.$ \end_inset The \begin_inset Quotes eld \end_inset factor \begin_inset Quotes erd \end_inset \begin_inset Formula $H/E$ \end_inset is thus \begin_inset Formula $-ik/\omega\mu=-i\sqrt{\varepsilon_{r}\varepsilon_{0}/\mu_{r}\mu_{0}}$ \end_inset , which is important in determining the Mie coefficients. \end_layout \begin_layout Standard The common multipole-dependent factor is given by \begin_inset Formula \[ E_{mn}=\left|E_{0}\right|i^{n}(2n+1)\frac{\left(n-m\right)!}{\left(n+m\right)!} \] \end_inset which \begin_inset Quotes eld \end_inset is desired for keeping the formulation of the multisphere scattering theory consistent with that of the Mie theory. It ensures that all the expressions in the multisphere theory turn out to be identical to those in the Mie theory when one is dealing with a cluster containing only one sphere and illuminated by a single plane wave \begin_inset Quotes erd \end_inset . (According to Bohren&Huffman \begin_inset CommandInset citation LatexCommand cite after "(4.37)" key "bohren_absorption_1983" \end_inset , the decomposition of a plane wave reads \begin_inset Formula \[ \vect E=E_{0}\sum_{n=1}^{\infty}i^{n}\frac{2n+1}{n(n+1)}\left(\vect M_{o1n}^{(1)}-i\vect N_{e1n}^{(1)}\right), \] \end_inset where the even/odd VSWF and \begin_inset Formula $m\ge0$ \end_inset convention is used.) \end_layout \begin_layout Standard \emph on It should be possible to just take it away and the abovementioned expansions are still consistent, are they not? \end_layout \begin_layout Standard In \begin_inset CommandInset citation LatexCommand cite after "sec. 4A" key "xu_electromagnetic_1995" \end_inset , there are formulae for translation of the plane wave between VSWF with different origins. \end_layout \begin_layout Standard o \end_layout \begin_layout Subsubsection Mie scattering \end_layout \begin_layout Standard For the exact form of the coefficients following from the boundary conditions on the spherical surface, cf. \begin_inset CommandInset citation LatexCommand cite after "(12–13)" key "xu_electromagnetic_1995" \end_inset . For the particular case of spherical nanoparticle, it is important that they can be written as \begin_inset CommandInset citation LatexCommand cite after "(14–15)" key "xu_electromagnetic_1995" \end_inset \begin_inset Formula \begin{alignat*}{1} a_{mn}^{j} & =R_{n}^{V}p_{mn}^{j},\quad b_{mn}^{j}=R_{n}^{H}q_{mn}^{j},\\ c_{mn}^{j} & =T_{n}^{H}q_{mn}^{j},\quad d_{mn}^{j}=T_{n}^{V}p_{mn}^{j}, \end{alignat*} \end_inset in other words, the Mie coefficients do not depend on \begin_inset Formula $m$ \end_inset but solely on \begin_inset Formula $n$ \end_inset (which is not surprising and probably follows from the Wigner-Eckart theorem). \end_layout \begin_layout Standard Respecting the conventions for decomposition in the previous section (i.e. there is opposite sign in the scattered part), the reflection and \begin_inset Quotes eld \end_inset transmission \begin_inset Quotes erd \end_inset coefficients become (adopted from \begin_inset CommandInset citation LatexCommand cite after "(4.52--53)" key "bohren_absorption_1983" \end_inset \begin_inset Formula \begin{eqnarray*} R_{n}^{V} & =\frac{a_{n}}{p_{n}}= & \frac{\mu_{e}m^{2}z^{i}ž^{e}-\mu_{i}z^{e}ž^{i}}{\mu_{e}m^{2}z^{i}ž^{s}-\mu_{i}z^{s}ž^{i}}\\ R_{n}^{H} & =\frac{b_{n}}{q_{n}}= & \frac{\mu_{i}z^{i}ž^{e}-\mu_{e}z^{e}ž^{i}}{\mu_{i}z^{i}ž^{s}-\mu_{e}z^{s}ž^{i}}\\ T_{n}^{V} & =\frac{d_{n}}{p_{n}}= & \frac{\mu_{i}mz^{e}ž^{s}-\mu_{i}mz^{s}ž^{e}}{\mu_{e}m^{2}z^{i}ž^{s}-\mu_{i}z^{s}ž^{i}}\\ T_{n}^{H} & =\frac{c_{n}}{q_{n}}= & \frac{\mu_{i}z^{e}ž^{s}-\mu_{i}z^{s}ž^{e}}{\mu_{i}z^{i}ž^{s}-\mu_{e}z^{s}ž^{i}} \end{eqnarray*} \end_inset where \begin_inset Formula $\mu_{i}|\mu_{e}$ \end_inset is (absolute) permeability of the sphere|envinronment, \begin_inset Formula $m=k_{i}/k_{e}=\sqrt{\mu_{i}\varepsilon_{i}/\mu_{e}\varepsilon_{e}}$ \end_inset , and \begin_inset Formula \begin{eqnarray*} z^{i} & = & z_{n}^{(J_{i}=1)}(k_{i}R)=j_{n}(k_{i}R),\\ z^{e} & = & z_{n}^{(J_{e})}(k_{e}R),\\ z^{s} & = & z_{n}^{(J_{s})}(k_{e}R),\\ ž^{i/e/s} & = & \frac{\ud(k_{i/e/e}R\cdot z_{n}^{(J_{i/e/e})}(k_{i/e/e}R)}{\ud(k_{i/e/e}R)}. \end{eqnarray*} \end_inset \end_layout \begin_layout Subsubsection Translation coefficients \end_layout \begin_layout Standard A quite detailed study can be found in \begin_inset CommandInset citation LatexCommand cite key "xu_calculation_1996" \end_inset , I have not read the recenter one \begin_inset CommandInset citation LatexCommand cite key "xu_efficient_1998" \end_inset which deals with efficient evaluation of Wigner 3jm symbols and Gaunt coefficie nts. \end_layout \begin_layout Standard With the VSWF as in \begin_inset CommandInset ref LatexCommand eqref reference "eq:vswf" \end_inset and translation relations in the form \begin_inset CommandInset citation LatexCommand cite after "(38,39)" key "xu_calculation_1996" \end_inset \begin_inset Formula \begin{eqnarray*} \vect M_{\mu\nu}^{(J)l} & = & \sum_{n=1}^{\infty}\sum_{m=-n}^{n}\left[A_{mn}^{\mu\nu}\vect M_{mn}^{(1)j}+B_{mn}^{\mu\nu}\vect N_{mn}^{(1)j}\right],\quad r\le d_{lj},\\ \vect N_{\mu\nu}^{(J)l} & = & \sum_{n=1}^{\infty}\sum_{m=-n}^{n}\left[B_{mn}^{\mu\nu}\vect M_{mn}^{(1)j}+A_{mn}^{\mu\nu}\vect N_{mn}^{(1)j}\right],\quad r\le d_{lj},\\ \vect M_{\mu\nu}^{(J)l} & = & \sum_{n=1}^{\infty}\sum_{m=-n}^{n}\left[A_{mn}^{\mu\nu}\vect M_{mn}^{(J)j}+B_{mn}^{\mu\nu}\vect N_{mn}^{(J)j}\right],\quad r\ge d_{lj},\\ \vect N_{\mu\nu}^{(J)l} & = & \sum_{n=1}^{\infty}\sum_{m=-n}^{n}\left[B_{mn}^{\mu\nu}\vect M_{mn}^{(J)j}+A_{mn}^{\mu\nu}\vect N_{mn}^{(J)j}\right],\quad r\ge d_{lj}, \end{eqnarray*} \end_inset the translation coefficients (which should in fact be also labeled with their origin indices \begin_inset Formula $l,j$ \end_inset ) are \begin_inset CommandInset citation LatexCommand cite after "(82,83)" key "xu_calculation_1996" \end_inset \end_layout \begin_layout Standard \begin_inset Formula \begin{multline*} A_{mn}^{\mu\nu}=\\ \frac{(-1)^{m}i^{\nu+n}(n+2)_{n-1}\left(\nu+2\right)_{\nu+1}(n+\nu+m-\mu)!}{4n(n+\nu+1)_{n+\nu}(n-m)!(\nu+m)!}\\ \times e^{i(\mu-m)\phi_{lj}}\sum_{q=0}^{q_{\mathrm{max}}}(-1)^{q}\left[n(n+1)+\nu(\nu+1)-p(p+1)\right]\\ \times\tilde{a}_{1q}\begin{pmatrix}z_{p}^{(J)}(kd_{lj})\\ j_{p}(kd_{lj}) \end{pmatrix}P_{p}^{\mu-m}(\cos\theta_{lj}),\qquad\begin{pmatrix}r\le d_{lj}\\ r\ge d_{lj} \end{pmatrix}; \end{multline*} \end_inset \begin_inset Formula \begin{multline*} B_{mn}^{\mu\nu}=\\ \frac{(-1)^{m}i^{\nu+n+1}(n+2)_{n+1}\left(\nu+2\right)_{\nu+1}(n+\nu+m-\mu+1)!}{4n(n+1)(n+m+1)(n+\nu+2)_{n+\nu+1}(n-m)!(\nu+m)!}\\ \times e^{i(\mu-m)\phi_{lj}}\sum_{q=0}^{Q_{\mathrm{max}}}(-1)^{q}\Big\{2(n+1)(\nu-\mu)\tilde{a}_{2q}-\\ -\left[p(p+3)-\nu(\nu+1)-n(n+3)-2\mu(n+1)\right]\tilde{a}_{3q}\Big\}\\ \times\begin{pmatrix}z_{p+1}^{(J)}(kd_{lj})\\ j_{p+1}(kd_{lj}) \end{pmatrix}P_{p+1}^{\mu-m}(\cos\theta_{lj}),\qquad\begin{pmatrix}r\le d_{lj}\\ r\ge d_{lj} \end{pmatrix}; \end{multline*} \end_inset where \begin_inset CommandInset citation LatexCommand cite after "(79,80)" key "xu_calculation_1996" \end_inset \begin_inset Formula \begin{eqnarray*} \tilde{a}_{1q} & = & a(-m,n,\mu,\nu,n+\nu-2q)/a(-m,n,\mu,\nu,n+\nu),\\ \tilde{a}_{2q} & = & a(-m-1,n+1,\mu+1,\nu,n+\nu+1-2q)/\\ & & /a(-m-1,n+1,\mu+1,\nu,n+\nu+1),\\ \tilde{a}_{3q} & = & a(-m,n+1,\mu,\nu,n+\nu+1-2q)/\\ & & /a(-m,n+1,\mu,\nu,\mu+\nu+1), \end{eqnarray*} \end_inset \begin_inset Formula \begin{eqnarray*} p & = & n+\nu-2q\\ q_{\max} & = & \min\left(n,\nu,\frac{n+\nu-\left|m-\mu\right|}{2}\right),\\ Q_{\max} & = & \min\left(n+1,\nu,\frac{n+\nu+1-\left|m-\mu\right|}{2}\right), \end{eqnarray*} \end_inset where the parentheses with lower index mean most likely the Pochhammer symbol / \emph on rising \emph default factorial \begin_inset Formula \[ \left(x\right)_{n}=x(x+1)(x+2)\dots(x+n-1)=\frac{(x+n-1)!}{(x-1)!}=\frac{\Gamma(x+n)}{\Gamma(x)}, \] \end_inset which is damn confusing (because this can also mean the falling factorial, cf. Wikipedia); and Xu does not bother explaining the notation \emph on anywhere \emph default . The fact that it is the rising factorial has been checked by comparing \begin_inset Formula $a_{0}$ \end_inset \begin_inset CommandInset citation LatexCommand cite after "(78)" key "xu_calculation_1996" \end_inset to some implementation from the internets \begin_inset Foot status open \begin_layout Plain Layout \family typewriter \begin_inset CommandInset href LatexCommand href name "https://raw.githubusercontent.com/michael-hartmann/gaunt/master/gaunt.py" target "https://raw.githubusercontent.com/michael-hartmann/gaunt/master/gaunt.py" \end_inset \end_layout \end_inset . \end_layout \begin_layout Standard The implementation should be checked with \begin_inset CommandInset citation LatexCommand cite after "Table II" key "xu_calculation_1996" \end_inset \end_layout \begin_layout Subsubsection Equations for the scattering problem \end_layout \begin_layout Standard The linear system for the scattering problem reads \begin_inset CommandInset citation LatexCommand cite after "(30)" key "xu_electromagnetic_1995" \end_inset \begin_inset Formula \begin{eqnarray*} a_{mn}^{j} & = & a_{n}^{j}\left\{ p_{mn}^{j,j}-\sum_{l\neq j}^{(1,L)}\sum_{\nu=1}^{\infty}\sum_{\mu=-\nu}^{\nu}\left[a_{\mu\nu}^{l}A_{mn}^{\mu\nu;lj}+b_{\mu\nu}^{l}B_{mn}^{\mu\nu;lj}\right]\right\} \\ b_{mn}^{j} & = & b_{n}^{j}\left\{ q_{mn}^{j,j}-\sum_{l\neq j}^{(1,L)}\sum_{\nu=1}^{\infty}\sum_{\mu=-\nu}^{\nu}\left[a_{\mu\nu}^{l}B_{mn}^{\mu\nu;lj}+b_{\mu\nu}^{l}A_{mn}^{\mu\nu;lj}\right]\right\} \end{eqnarray*} \end_inset where \begin_inset Formula $p_{mn}^{j,j},q_{mn}^{j,j}$ \end_inset are the expansion coefficients of the initial incident waves in the \begin_inset Formula $j$ \end_inset -th particle's coordinate system \begin_inset CommandInset citation LatexCommand cite after "sec. 4A" key "xu_electromagnetic_1995" \end_inset . \emph on TODO expressions for \begin_inset Formula $p_{mn}^{j,j},q_{mn}^{j,j}$ \end_inset in the case of dipole initial wave. \end_layout \begin_layout Subsubsection Solving the linear system \end_layout \begin_layout Standard \begin_inset CommandInset citation LatexCommand cite after "sec. 5" key "xu_electromagnetic_1995" \end_inset \end_layout \begin_layout Subsection T-Matrix resummation (multiple scatterers) \end_layout \begin_layout Subsection Boundary element method \end_layout \begin_layout Subsection BEM→TM \end_layout \begin_layout Standard Cf. SCUFF-TMATRIX ( \begin_inset CommandInset ref LatexCommand ref reference "sub:SCUFF-TMATRIX" \end_inset ) \end_layout \begin_layout Section Available software \end_layout \begin_layout Itemize TODO which of them can calculate the VSWF translation coefficients? \end_layout \begin_layout Subsection SCUFF-EM \begin_inset CommandInset citation LatexCommand cite key "reid_scuff-em_2015" \end_inset \end_layout \begin_layout Subsubsection \family typewriter SCUFF-TMATRIX \family default \begin_inset CommandInset label LatexCommand label name "sub:SCUFF-TMATRIX" \end_inset \end_layout \begin_layout Subsubsection \family typewriter SCUFF-SCATTER \family default \begin_inset CommandInset label LatexCommand label name "sub:SCUFF-SCATTER" \end_inset \end_layout \begin_layout Subsubsection Caveats \end_layout \begin_layout Description Units. \family typewriter SCUFF-SCATTER \family default 's Angular frequencies specified using the \family typewriter --Omega \family default or \family typewriter --OmegaFile \family default arguments are interpreted in units of \begin_inset Formula $c/1\,\mathrm{μm}=3\cdot10^{14}\,\mathrm{rad/s}$ \end_inset \begin_inset Foot status open \begin_layout Plain Layout \family typewriter \begin_inset CommandInset href LatexCommand href name "http://homerreid.dyndns.org/scuff-EM/scuff-scatter/scuffScatterExamples.shtml" target "http://homerreid.dyndns.org/scuff-EM/scuff-scatter/scuffScatterExamples.shtml" \end_inset \end_layout \end_inset . \emph on TODO what are the output units? \end_layout \begin_layout Subsection MSTM \begin_inset CommandInset citation LatexCommand cite key "mackowski_mstm_2013" \end_inset \end_layout \begin_layout Itemize The incident field is a gaussian beam or a plane wave in the vanilla code (no multipole radiation as input!). \end_layout \begin_layout Itemize The bulk of the useful code is in the \family typewriter mstm-modules-v3.0.f90 \family default file. \end_layout \begin_layout Itemize For solving the interaction equations \begin_inset CommandInset citation LatexCommand cite after "(14)" key "mackowski_mstm_2013" \end_inset , the BCGM (biconjugate gradient method) is used. (According to Wikipedia, this method is numerically unstable but has a stabilized version (stabilized BCGM).) \end_layout \begin_layout Itemize According to the manual \begin_inset CommandInset citation LatexCommand cite after "2.3" key "mackowski_mstm_2013" \end_inset , they use some method (rotational-axial translation decomposition of the translation operation), which \begin_inset Quotes eld \end_inset reduces the operation from an \begin_inset Formula $L_{S}^{4}$ \end_inset process to \begin_inset Formula $L_{S}^{3}$ \end_inset process where \begin_inset Formula $L_{S}$ \end_inset is the truncation order of the expansion \begin_inset Quotes erd \end_inset (more details can probably be found at \begin_inset CommandInset citation LatexCommand cite after "around (68)" key "mackowski_calculation_1996" \end_inset . \end_layout \begin_deeper \begin_layout Itemize \emph on Not sure if this holds also for nonspherical particles, I should either read carefully \emph default \begin_inset CommandInset citation LatexCommand cite key "mackowski_calculation_1996" \end_inset \emph on or look into \begin_inset CommandInset citation LatexCommand cite key "mishchenko_electromagnetic_2003" \end_inset which is also cited in the manual. \end_layout \end_deeper \begin_layout Itemize By default spheres, it is possible to add own T-Matrix coefficients instead. \end_layout \begin_deeper \begin_layout Itemize \emph on Is it then possible to insert a T-Matrix of an arbitrary shape, or is it somehow limited to \begin_inset Quotes eld \end_inset spherical-like \begin_inset Quotes erd \end_inset particles? \end_layout \end_deeper \begin_layout Itemize Why the heck are the T-matrix options listed in the \begin_inset Quotes eld \end_inset Options for random orientation calculations \begin_inset Quotes erd \end_inset ? Well, it seems that for fixed orientation, it is not possible to specify the T-matrix, cf. the description of \family typewriter fixed_or_random_orientation \family default option in \begin_inset CommandInset citation LatexCommand cite after "3.2.3" key "mackowski_mstm_2013" \end_inset . \end_layout \begin_layout Subsubsection Interesting subroutines \end_layout \begin_layout Itemize \family typewriter rottranfarfield \family default : it states \begin_inset Quotes eld \end_inset far field formula for outgoing vswf translation \begin_inset Quotes erd \end_inset . What is that and how does it differ from whatever else (near field?) formula? \end_layout \begin_layout Subsection py_gmm \begin_inset CommandInset citation LatexCommand cite key "pellegrini_py_gmm_2015" \end_inset \end_layout \begin_layout Itemize Fortran code, already (partially) pythonized using \family typewriter f2py \family default by the authors(?); under GNU GPLv3. This could save my day. \end_layout \begin_layout Itemize Lots of unnecessary code duplication (see e.g. \family typewriter coeff_sp2 \family default and \family typewriter coeff_sp2_dip \family default subroutines). \end_layout \begin_layout Itemize Has comments!!! (Sometimes they are slightly inaccurate due to the copy-pasting, but it is still one of the most readable FORTRAN codes I have seen.) \end_layout \begin_layout Itemize The subroutines seem not to be bloated with dependencies on static/global variables, so they should be quite easily reusable. \end_layout \begin_layout Itemize The FORTRAN code was apparently used in \begin_inset CommandInset citation LatexCommand cite key "pellegrini_interacting_2007" \end_inset . Uses the multiple-scattering formalism described in \begin_inset CommandInset citation LatexCommand cite key "xu_efficient_1998" \end_inset . \end_layout \begin_layout Subsubsection Interesting subroutines \end_layout \begin_layout Standard Mie scattering: \end_layout \begin_layout Itemize \family typewriter coeff_sp2 \family default : calculation of the Mie scattering coefficients ( \begin_inset Formula $\overline{a}_{n}^{l},\overline{b}_{n}^{l}$ \end_inset as in \begin_inset CommandInset citation LatexCommand cite after "(1), (2), \\ldots" key "pellegrini_py_gmm_2015" \end_inset ), for a set of spheres (therefore all the parameters have +1 dimension). \end_layout \begin_deeper \begin_layout Itemize What does the input parameter \family typewriter v_req \family default ( \emph on vettore raggi equivalenti \emph default ) mean? \end_layout \begin_layout Itemize How do I put in the environment permittivity? \end_layout \begin_layout Itemize \family typewriter m_epseq \family default are real and imaginary parts of the permittivity (which are then transformed into complex \family typewriter v_epsc \family default ) \end_layout \begin_layout Itemize \family typewriter ref_index \family default is the environment refractive index (called \family typewriter n_matrix \family default in the example ipython notebook) \end_layout \begin_layout Itemize \family typewriter v_req \family default are the sphere radii? \end_layout \begin_layout Itemize \family typewriter nstop \family default is the maximum order of the \begin_inset Formula $n$ \end_inset -expansion \end_layout \begin_layout Itemize \family typewriter neq \family default is ns, number of spheres for which the calculation is performed apparently, it is connected to some \begin_inset Quotes eld \end_inset dirty hack to interface fortran and python properly \begin_inset Quotes erd \end_inset (cf. \family typewriter gmm_f2py_module.f90 \family default ) \end_layout \end_deeper \begin_layout Section Implementation / code integration \end_layout \begin_layout Standard There are several scipy functions to compute the Legendre polynomials. lpmv is ufunc, whereas lpmn is not; lpmn can, however, compute also the derivatives. This is a bit annoying, because I have to iterate the positions with a for loop. \end_layout \begin_layout Standard The default gsl legendre function (gsl_sf_legendre_array) without additional parameters has opposite sign than the scipy.special.lpmn, and it should contain the Condon-Shortley phase; thus scipy.special.lpmn probably does NOT include the CS phase. But again, this should hopefully play no role. The overall normalisation, on the other hand, plays huge role. \end_layout \begin_layout Subsection Scattering-Taylor.ipynb \end_layout \begin_layout Standard In the conventions used in the code and the corresponding libraries, the following symmetries hold for \begin_inset Formula $J=1$ \end_inset (regular wavefunctions): \begin_inset Formula \begin{eqnarray*} \widetilde{\vect M}_{m,n}^{(1)} & = & (-1)^{m}\widetilde{\vect M}_{-m,n}^{(1)},\\ \widetilde{\vect N}_{m,n}^{(1)} & = & (-1)^{m}\widetilde{\vect N}_{-m,n}^{(1)}. \end{eqnarray*} \end_inset \end_layout \begin_layout Section Testing and reproduction of foreign results \end_layout \begin_layout Subsection Delga PRL \begin_inset CommandInset citation LatexCommand cite key "delga_quantum_2014" \end_inset \end_layout \begin_layout Subsubsection Parameters \end_layout \begin_layout Itemize Surrounding lossless dielectric \series bold medium \series default with permittivity \begin_inset Formula $\epsilon_{d}=2.13$ \end_inset . \end_layout \begin_layout Itemize \series bold QEs: \series default dipole moment \begin_inset Formula $\mu=0.19\, e\cdot\mathrm{nm}=9.12\,\mathrm{D}$ \end_inset , count \begin_inset Formula $N\in\left\{ 1,50,100,200\right\} $ \end_inset , radial orientation, \begin_inset Formula $h=1\,\mathrm{nm}$ \end_inset above the sphere (except for Fig. 5 where variable), natural frequency \begin_inset Formula $\Omega_{n}=\omega_{0}-i\gamma_{\mathrm{QE}}/2,$ \end_inset \begin_inset Formula $\omega_{0}=$ \end_inset varies, \begin_inset Formula $\gamma_{\mathrm{QE}}=15\,\mathrm{meV}$ \end_inset . \end_layout \begin_layout Itemize \series bold Sphere: \end_layout \begin_deeper \begin_layout Itemize radius \begin_inset Formula $a=7\,\mathrm{nm}$ \end_inset , \end_layout \begin_layout Itemize Drude model \begin_inset Formula $\epsilon_{m}(\omega)=\epsilon_{\infty}-\frac{\omega_{p}^{2}}{\omega\left(\omega+i\gamma_{p}\right)}$ \end_inset \end_layout \begin_deeper \begin_layout Itemize Drude parameters \begin_inset Formula $\omega_{p}=9\,\mathrm{eV}$ \end_inset , \begin_inset Formula $\epsilon_{\infty}=4.6$ \end_inset , \begin_inset Formula $\gamma_{p}=0.1\,\mathrm{eV}$ \end_inset \end_layout \end_deeper \begin_layout Itemize background permittivity \begin_inset Formula $\epsilon_{d}(\omega)=2.13$ \end_inset \end_layout \begin_layout Itemize (approximate?; not really a parameter) LSP resonances \begin_inset Formula $\omega_{l}=\omega_{p}/\sqrt{\epsilon_{\infty}+\left(1+1/l\right)\epsilon_{d}}$ \end_inset ; particularly, \begin_inset Formula $\omega_{1}\approx3.0236\,\mathrm{eV}$ \end_inset , \begin_inset Formula $\omega_{2}\approx3.2236\,\mathrm{eV}$ \end_inset , \begin_inset Formula $\omega_{3}\approx3.30\,\mathrm{eV}$ \end_inset , \begin_inset Formula $\omega_{4}\approx3.34\,\mathrm{eV}$ \end_inset , \begin_inset Formula $\omega_{5}\approx3.364\,\mathrm{eV}$ \end_inset \begin_inset Formula $\omega_{\infty}\approx3.4692\,\mathrm{eV}$ \end_inset \end_layout \end_deeper \begin_layout Itemize \series bold Detector: \series default \end_layout \begin_deeper \begin_layout Itemize Far field: \begin_inset Formula $1\,\mathrm{\mu m}$ \end_inset away from the center of the nanoparticle along the \begin_inset Formula $y$ \end_inset axis (Fig. 3). \end_layout \begin_layout Itemize Near field: position not specified in the paper; but in Fig. 4(b) there are \begin_inset Quotes eld \end_inset polarization spectra \begin_inset Quotes erd \end_inset instead of \begin_inset Quotes eld \end_inset light spectra \begin_inset Quotes erd \end_inset (eq. 4) in Fig. 4(a). Does this mean that they are evaluated somewhere in/on the sphere? Or in the particle? The latter is likely, as it is given by \begin_inset Formula $P_{n}\left(\omega\right)=\left\langle \sigma_{n}^{+}\left(-\omega\right)\sigma_{n}^{-}(\omega)\right\rangle $ \end_inset (cf. the column below Fig. 3). \end_layout \end_deeper \begin_layout Subsubsection Testing \end_layout \begin_layout Standard In my \begin_inset Quotes eld \end_inset old \begin_inset Quotes erd \end_inset code, there no splitting observable around \begin_inset Formula $\omega\approx\omega_{0}\approx\omega_{\infty}\approx3.46\,\mathrm{eV}$ \end_inset . This is perhaps because the couplings to the higher multipoles is miscalculated (too small). No splitting around the NP dipole ( \begin_inset Formula $\approx3,02\,\mathrm{eV}$ \end_inset ) should be OK for single QE in far field (cf. Fig. 3). And there are yet the \begin_inset Quotes eld \end_inset switched axes \begin_inset Quotes erd \end_inset ... \end_layout \begin_layout Standard If I set the dipole reflection coefficients RH[1], RV[1] to zero, and multiply the the quadrupole reflection coefficients RH[2], RV[2] by \begin_inset Formula $10^{6}$ \end_inset , the peak at \begin_inset Formula $3.0\,\mathrm{eV}$ \end_inset dissapears and a tiny(!) peak appears around the (expected) position of \begin_inset Formula $3.0\,\mathrm{eV}$ \end_inset . Have I fucked up the Mie reflection coefficients? Sounds like if I forgot a factor of \begin_inset Formula $c$ \end_inset somewhere. \end_layout \begin_layout Subsection Delga JoO \begin_inset CommandInset citation LatexCommand cite key "delga_theory_2014" \end_inset \end_layout \begin_layout Subsubsection Parameters \end_layout \begin_layout Itemize \series bold QEs: \series default dipole moment \begin_inset Formula $\mu=0.38\, e\cdot\mathrm{nm}=18.24\,\mathrm{D}$ \end_inset (double), otherwise the same parameters as in \begin_inset CommandInset citation LatexCommand cite key "delga_quantum_2014" \end_inset . \end_layout \begin_layout Itemize \series bold Sphere: \series default as in \begin_inset CommandInset citation LatexCommand cite key "delga_quantum_2014" \end_inset \end_layout \begin_layout Itemize \series bold Detector: \series default not stated in the paper \end_layout \begin_layout Itemize \series bold Numerics: \series default looking at the leftmost ball in Fig. 3, it seems that their SVW cutoff is around 12. \end_layout \begin_layout Section Misc \end_layout \begin_layout Itemize The \begin_inset Quotes eld \end_inset zero limits \begin_inset Quotes erd \end_inset of \begin_inset Formula $\tilde{\pi},\tilde{\tau}$ \end_inset functions in Taylor's normalisation can be expressed as \lang finnish \begin_inset Formula \begin{eqnarray*} \lim_{\theta\to0}\tilde{\pi}_{mn}\left(\cos\theta\right) & = & -\frac{1}{2}\sqrt{\frac{2n+1}{4\pi}n\left(n+1\right)}(\delta_{m,1}+\delta_{m,-1})\\ \lim_{\theta\to0}\tilde{\tau}_{mn}\left(\cos\theta\right) & = & -\frac{1}{2}\sqrt{\frac{2n+1}{4\pi}n\left(n+1\right)}(\delta_{m,1}-\delta_{m,-1}) \end{eqnarray*} \end_inset \end_layout \begin_layout Standard \begin_inset CommandInset bibtex LatexCommand bibtex bibfiles "dipdip" options "apsrev" \end_inset \end_layout \end_body \end_document