#LyX 2.4 created this file. For more info see https://www.lyx.org/ \lyxformat 583 \begin_document \begin_header \save_transient_properties true \origin unavailable \textclass article \use_default_options true \maintain_unincluded_children false \language english \language_package default \inputencoding utf8 \fontencoding auto \font_roman "default" "TeX Gyre Pagella" \font_sans "default" "default" \font_typewriter "default" "default" \font_math "auto" "auto" \font_default_family default \use_non_tex_fonts false \font_sc false \font_roman_osf true \font_sans_osf false \font_typewriter_osf false \font_sf_scale 100 100 \font_tt_scale 100 100 \use_microtype false \use_dash_ligatures true \graphics default \default_output_format default \output_sync 0 \bibtex_command default \index_command default \float_placement class \float_alignment class \paperfontsize default \spacing single \use_hyperref true \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 default \use_geometry false \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 \use_minted 0 \use_lineno 0 \index Index \shortcut idx \color #008000 \end_index \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \paragraph_indentation default \is_math_indent 0 \math_numbering_side default \quotes_style english \dynamic_quotes 0 \papercolumns 1 \papersides 1 \paperpagestyle default \tablestyle 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 Section Finite systems \end_layout \begin_layout Itemize motivation (classes of problems that this can solve: response to external radiation, resonances, ...) \begin_inset Separator latexpar \end_inset \end_layout \begin_deeper \begin_layout Itemize theory \begin_inset Separator latexpar \end_inset \end_layout \begin_deeper \begin_layout Itemize T-matrix definition, basics \begin_inset Separator latexpar \end_inset \end_layout \begin_deeper \begin_layout Itemize How to get it? \end_layout \end_deeper \begin_layout Itemize translation operators (TODO think about how explicit this should be, but I guess it might be useful to write them to write them explicitly (but in the shortest possible form) in the normalisation used in my program) \end_layout \begin_layout Itemize employing point group symmetries and decomposing the problem to decrease the computational complexity (maybe separately) \end_layout \end_deeper \end_deeper \begin_layout Subsection Motivation \end_layout \begin_layout Standard The basic idea of MSTMM is quite simple: the driving electromagnetic field incident onto a scatterer is expanded into a vector spherical wavefunction (VSWF) basis in which the single scattering problem is solved, and the scattered field is then re-expanded into VSWFs centered at the other scatterers. Repeating the same procedure with all (pairs of) scatterers yields a set of linear equations, solution of which gives the coefficients of the scattered field in the VSWF bases. Once these coefficients have been found, one can evaluate various quantities related to the scattering (such as cross sections or the scattered fields) quite easily. \end_layout \begin_layout Standard However, the expressions appearing in the re-expansions are fairly complicated, and the implementation of MSTMM is extremely error-prone also due to the various conventions used in the literature. Therefore although we do not re-derive from scratch the expressions that can be found elsewhere in literature, we always state them explicitly in our convention. \end_layout \begin_layout Subsection Single-particle scattering \end_layout \begin_layout Standard In order to define the basic concepts, let us first consider the case of EM radiation scattered by a single particle. We assume that the scatterer lies inside a closed sphere \begin_inset Formula $\particle$ \end_inset , the space outside this volume \begin_inset Formula $\medium$ \end_inset is filled with an homogeneous isotropic medium with relative electric permittiv ity \begin_inset Formula $\epsilon(\vect r,\omega)=\epsbg(\omega)$ \end_inset and magnetic permeability \begin_inset Formula $\mu(\vect r,\omega)=\mubg(\omega)$ \end_inset , and that the whole system is linear, i.e. the material properties of neither the medium nor the scatterer depend on field intensities. Under these assumptions, the EM fields \begin_inset Formula $\vect{\Psi}=\vect E,\vect H$ \end_inset in \begin_inset Formula $\medium$ \end_inset must satisfy the homogeneous vector Helmholtz equation together with the transversality condition \begin_inset Formula \begin{equation} \left(\nabla^{2}+k^{2}\right)\Psi\left(\vect r,\vect{\omega}\right)=0,\quad\nabla\cdot\vect{\Psi}\left(\vect r,\vect{\omega}\right)=0\label{eq:Helmholtz eq} \end{equation} \end_inset \begin_inset Note Note status open \begin_layout Plain Layout frequency-space Maxwell's equations \begin_inset Formula \begin{align*} \nabla\times\vect E\left(\vect r,\omega\right)-ik\eta_{0}\eta\vect H\left(\vect r,\omega\right) & =0,\\ \eta_{0}\eta\nabla\times\vect H\left(\vect r,\omega\right)+ik\vect E\left(\vect r,\omega\right) & =0. \end{align*} \end_inset \end_layout \end_inset \begin_inset Note Note status open \begin_layout Plain Layout todo define \begin_inset Formula $\Psi$ \end_inset , mention transversality \end_layout \end_inset with \begin_inset Formula $k=k\left(\omega\right)=\omega\sqrt{\mubg(\omega)\epsbg(\omega)}/c_{0}$ \end_inset , as can be derived from the Maxwell's equations [REF Jackson?]. \end_layout \begin_layout Subsubsection Spherical waves \end_layout \begin_layout Standard Equation \begin_inset CommandInset ref LatexCommand ref reference "eq:Helmholtz eq" plural "false" caps "false" noprefix "false" \end_inset can be solved by separation of variables in spherical coordinates to give the solutions – the \emph on regular \emph default and \emph on outgoing \emph default vector spherical wavefunctions (VSWFs) \begin_inset Formula $\vswfrtlm{\tau}lm\left(k\vect r\right)$ \end_inset and \begin_inset Formula $\vswfouttlm{\tau}lm\left(k\vect r\right)$ \end_inset , respectively, defined as follows: \begin_inset Formula \begin{align*} \vswfrtlm 1lm\left(k\vect r\right) & =j_{l}\left(kr\right)\vsh 1lm\left(\uvec r\right),\\ \vswfrtlm 2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krj_{l}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(kr\right)}{kr}\vsh 3lm\left(\uvec r\right), \end{align*} \end_inset \begin_inset Formula \begin{align*} \vswfouttlm 1lm\left(k\vect r\right) & =h_{l}^{\left(1\right)}\left(kr\right)\vsh 1lm\left(\uvec r\right),\\ \vswfouttlm 2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krh_{l}^{\left(1\right)}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{h_{l}^{\left(1\right)}\left(kr\right)}{kr}\vsh 3lm\left(\uvec r\right),\\ & \tau=1,2;\quad l=1,2,3,\dots;\quad m=-l,-l+1,\dots,+l, \end{align*} \end_inset where \begin_inset Formula $\vect r=r\uvec r$ \end_inset , \begin_inset Formula $j_{l}\left(x\right),h_{l}^{\left(1\right)}\left(x\right)=j_{l}\left(x\right)+iy_{l}\left(x\right)$ \end_inset are the regular spherical Bessel function and spherical Hankel function of the first kind, respectively, as in [DLMF §10.47], and \begin_inset Formula $\vsh{\tau}lm$ \end_inset are the \emph on vector spherical harmonics \emph default \begin_inset Formula \begin{align*} \vsh 1lm & =\\ \vsh 2lm & =\\ \vsh 3lm & = \end{align*} \end_inset In our convention, the (scalar) spherical harmonics \begin_inset Formula $\ush lm$ \end_inset are identical to those in [DLMF 14.30.1], i.e. \begin_inset Formula \[ \ush lm=\left(\frac{\left(l-m\right)!\left(2l+1\right)}{4\pi\left(l+m\right)!}\right)^{\frac{1}{2}}e^{im\phi}\dlmfFer lm\left(\cos\theta\right) \] \end_inset where importantly, the Ferrers functions \begin_inset Formula $\dlmfFer lm$ \end_inset defined as in [DLMF §14.3(i)] do already contain the Condon-Shortley phase \begin_inset Formula $\left(-1\right)^{m}$ \end_inset . \begin_inset Note Note status open \begin_layout Plain Layout TODO názornější definice. \end_layout \end_inset \end_layout \begin_layout Standard The convention for VSWFs used here is the same as in [Kristensson 2014]; over other conventions used elsewhere in literature, it has several fundamental advantages – most importantly, the translation operators introduced later in eq. \begin_inset CommandInset ref LatexCommand ref reference "eq:translation op def" plural "false" caps "false" noprefix "false" \end_inset are unitary, and it gives the simplest possible expressions for power transport and cross sections without additional \begin_inset Formula $l,m$ \end_inset -dependent factors (for that reason, we also call our VSWFs as \emph on power-normalised \emph default ). Power-normalisation and unitary translation operators are possible to achieve also with real spherical harmonics – such a convention is used in \begin_inset CommandInset citation LatexCommand cite key "kristensson_scattering_2016" literal "false" \end_inset . \end_layout \begin_layout Standard \begin_inset Note Note status open \begin_layout Plain Layout Its solutions (TODO under which conditions? What vector space do the SVWFs actually span? Check Comment 9.2 and Appendix f.9.1 in Kristensson) \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Note Note status open \begin_layout Plain Layout TODO small note about cartesian multipoles, anapoles etc. (There should be some comparing paper that the Russians at META 2018 mentioned.) \end_layout \end_inset \end_layout \begin_layout Subsubsection T-matrix definition \end_layout \begin_layout Standard The regular VSWFs \begin_inset Formula $\vswfrtlm{\tau}lm\left(k\vect r\right)$ \end_inset constitute a basis for solutions of the Helmholtz equation \begin_inset CommandInset ref LatexCommand ref reference "eq:Helmholtz eq" plural "false" caps "false" noprefix "false" \end_inset inside a ball \begin_inset Formula $\openball 0R$ \end_inset with radius \begin_inset Formula $R$ \end_inset and center in the origin; however, if the equation is not guaranteed to hold inside a smaller ball \begin_inset Formula $B_{0}\left(R_{<}\right)$ \end_inset around the origin (typically due to presence of a scatterer), one has to add the outgoing VSWFs \begin_inset Formula $\vswfrtlm{\tau}lm\left(k\vect r\right)$ \end_inset to have a complete basis of the solutions in the volume \begin_inset Formula $\openball 0R\backslash B_{0}\left(R_{<}\right)$ \end_inset . \begin_inset Note Note status open \begin_layout Plain Layout Vnitřní koule uzavřená? Jak se řekne mezikulí anglicky? \end_layout \end_inset \end_layout \begin_layout Standard The single-particle scattering problem at frequency \begin_inset Formula $\omega$ \end_inset can be posed as follows: Let a scatterer be enclosed inside the ball \begin_inset Formula $B_{0}\left(R_{<}\right)$ \end_inset and let the whole volume \begin_inset Formula $\openball 0R\backslash B_{0}\left(R_{<}\right)$ \end_inset be filled with a homogeneous isotropic medium with wave number \begin_inset Formula $k\left(\omega\right)$ \end_inset . Inside this volume, the electric field can be expanded as \begin_inset Note Note status open \begin_layout Plain Layout doplnit frekvence a polohy \end_layout \end_inset \begin_inset Formula \begin{equation} \vect E\left(\omega,\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\left(\rcoefftlm{\tau}lm\vswfrtlm{\tau}lm+\outcoefftlm{\tau}lm\vswfouttlm{\tau}lm\right).\label{eq:E field expansion} \end{equation} \end_inset If there was no scatterer and \begin_inset Formula $B_{0}\left(R_{<}\right)$ \end_inset was filled with the same homogeneous medium, the part with the outgoing VSWFs would vanish and only the part \begin_inset Formula $\vect E_{\mathrm{inc}}=\sum_{\tau lm}\rcoefftlm{\tau}lm\vswfrtlm{\tau}lm$ \end_inset due to sources outside \begin_inset Formula $\openball 0R$ \end_inset would remain. Let us assume that the \begin_inset Quotes eld \end_inset driving field \begin_inset Quotes erd \end_inset is given, so that presence of the scatterer does not affect \begin_inset Formula $\vect E_{\mathrm{inc}}$ \end_inset and is fully manifested in the latter part, \begin_inset Formula $\vect E_{\mathrm{scat}}=\sum_{\tau lm}\outcoefftlm{\tau}lm\vswfouttlm{\tau}lm$ \end_inset . We also assume that the scatterer is made of optically linear materials, and hence reacts on the incident field in a linear manner. This gives a linearity constraint between the expansion coefficients \begin_inset Formula \begin{equation} \outcoefftlm{\tau}lm=\sum_{\tau'l'm'}T_{\tau lm}^{\tau'l'm'}\rcoefftlm{\tau'}{l'}{m'}\label{eq:T-matrix definition} \end{equation} \end_inset where the \begin_inset Formula $T_{\tau lm}^{\tau'l'm'}=T_{\tau lm}^{\tau'l'm'}\left(\omega\right)$ \end_inset are the elements of the \emph on transition matrix, \emph default a.k.a. \begin_inset Formula $T$ \end_inset -matrix. It completely describes the scattering properties of a linear scatterer, so with the knowledge of the \begin_inset Formula $T$ \end_inset -matrix, we can solve the single-patricle scatering prroblem simply by substitut ing appropriate expansion coefficients \begin_inset Formula $\rcoefftlm{\tau'}{l'}{m'}$ \end_inset of the driving field into \begin_inset CommandInset ref LatexCommand ref reference "eq:T-matrix definition" plural "false" caps "false" noprefix "false" \end_inset . \end_layout \begin_layout Standard \begin_inset Note Note status open \begin_layout Plain Layout TOOD H-field expansion here? \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Formula $T$ \end_inset -matrices of particles with certain simple geometries (most famously spherical) can be obtained analytically [Kristensson 2016, Mie], but in general one can find them numerically by simulating scattering of a regular spherical wave \begin_inset Formula $\vswfouttlm{\tau}lm$ \end_inset and projecting the scattered fields (or induced currents, depending on the method) onto the outgoing VSWFs \begin_inset Formula $\vswfrtlm{\tau}{'l'}{m'}$ \end_inset . In practice, one can compute only a finite number of elements with a cut-off value \begin_inset Formula $L$ \end_inset on the multipole degree, \begin_inset Formula $l,l'\le L$ \end_inset , see below. We typically use the scuff-tmatrix tool from the free software SCUFF-EM suite [SCUFF-EM]. Note that older versions of SCUFF-EM contained a bug that rendered almost all \begin_inset Formula $T$ \end_inset -matrix results wrong; we found and fixed the bug and from upstream version xxx onwards, it should behave correctly. \end_layout \begin_layout Subsubsection T-matrix compactness, cutoff validity \end_layout \begin_layout Standard The magnitude of the \begin_inset Formula $T$ \end_inset -matrix elements depends heavily on the scatterer's size compared to the wavelength. Fortunately, the \begin_inset Formula $T$ \end_inset -matrix of a bounded scatterer is a compact operator [REF???], so from certain multipole degree onwards, \begin_inset Formula $l,l'>L$ \end_inset , the elements of the \begin_inset Formula $T$ \end_inset -matrix are negligible, so truncating the \begin_inset Formula $T$ \end_inset -matrix at finite multipole degree \begin_inset Formula $L$ \end_inset gives a good approximation of the actual infinite-dimensional itself. If the incident field is well-behaved, i.e. the expansion coefficients \begin_inset Formula $\rcoefftlm{\tau'}{l'}{m'}$ \end_inset do not take excessive values for \begin_inset Formula $l'>L$ \end_inset , the scattered field expansion coefficients \begin_inset Formula $\outcoefftlm{\tau}lm$ \end_inset with \begin_inset Formula $l>L$ \end_inset will also be negligible. \begin_inset Note Note status open \begin_layout Plain Layout TODO when it will not be negligible \end_layout \end_inset \end_layout \begin_layout Standard A rule of thumb to choose the \begin_inset Formula $L$ \end_inset with desired \begin_inset Formula $T$ \end_inset -matrix element accuracy \begin_inset Formula $\delta$ \end_inset can be obtained from the spherical Bessel function expansion around zero, TODO. \end_layout \begin_layout Subsubsection Power transport \end_layout \begin_layout Standard For convenience, let us introduce a short-hand matrix notation for the expansion coefficients and related quantities, so that we do not need to write the indices explicitly; so for example, eq. \begin_inset CommandInset ref LatexCommand ref reference "eq:T-matrix definition" plural "false" caps "false" noprefix "false" \end_inset would be written as \begin_inset Formula $\outcoeffp{}=T\rcoeffp{}$ \end_inset , where \begin_inset Formula $\rcoeffp{},\outcoeffp{}$ \end_inset are column vectors with the expansion coefficients. Transposed and complex-conjugated matrices are labeled with the \begin_inset Formula $\dagger$ \end_inset superscript. \end_layout \begin_layout Standard With this notation, we state an important result about power transport, derivation of which can be found in \begin_inset CommandInset citation LatexCommand cite after "sect. 7.3" key "kristensson_scattering_2016" literal "true" \end_inset . Let the field in \begin_inset Formula $\openball 0R\backslash B_{0}\left(R_{<}\right)$ \end_inset have expansion as in \begin_inset CommandInset ref LatexCommand ref reference "eq:E field expansion" plural "false" caps "false" noprefix "false" \end_inset . Then the net power transported from \begin_inset Formula $B_{0}\left(R_{<}\right)$ \end_inset to \begin_inset Formula $\openball 0R\backslash B_{0}\left(R_{<}\right)$ \end_inset via by electromagnetic radiation is \begin_inset Formula \begin{equation} P=\frac{1}{2k^{2}\eta_{0}\eta}\left(\Re\left(\rcoeffp{}^{\dagger}\outcoeffp{}\right)+\left\Vert \outcoeffp{}\right\Vert ^{2}\right)=\frac{1}{2k^{2}\eta_{0}\eta}\rcoeffp{}^{\dagger}\left(\Tp{}^{\dagger}\Tp{}+\frac{\Tp{}^{\dagger}+\Tp{}}{2}\right)\rcoeffp{}.\label{eq:Power transport} \end{equation} \end_inset In realistic scattering setups, power is transferred by radiation into \begin_inset Formula $B_{0}\left(R_{<}\right)$ \end_inset and absorbed by the enclosed scatterer, so \begin_inset Formula $P$ \end_inset is negative and its magnitude equals to power absorbed by the scatterer. \end_layout \begin_layout Subsubsection Plane wave expansion \end_layout \begin_layout Standard In many scattering problems considered in practice, the driving field is a plane wave. A transversal ( \begin_inset Formula $\vect k\cdot\vect E_{0}=0$ \end_inset ) plane wave propagating in direction \begin_inset Formula $\uvec k$ \end_inset with (complex) amplitude \begin_inset Formula $\vect E_{0}$ \end_inset can be expanded into regular VSWFs [REF KRIS] as \begin_inset Formula \[ \vect E_{\mathrm{PW}}\left(\vect r,\omega\right)=\vect E_{0}e^{ik\uvec k\cdot\vect r}=\sum_{\tau,l,m}\rcoeffptlm{}{\tau}lm\left(\vect k,\vect E_{0}\right)\vswfrtlm{\tau}lm\left(k\vect r\right), \] \end_inset with expansion coefficients \begin_inset Formula \begin{eqnarray} \rcoeffptlm{}1lm\left(\vect k,\vect E_{0}\right) & = & 4\pi i^{l}\vshD 1lm\left(\uvec k\right),\nonumber \\ \rcoeffptlm{}2lm\left(\vect k,\vect E_{0}\right) & = & -4\pi i^{l+1}\vshD 2lm\left(\uvec k\right).\label{eq:plane wave expansion} \end{eqnarray} \end_inset where \begin_inset Formula $\vshD{\tau}lm$ \end_inset are the \begin_inset Quotes eld \end_inset dual \begin_inset Quotes erd \end_inset vector spherical harmonics defined by duality relation with the \begin_inset Quotes eld \end_inset usual \begin_inset Quotes erd \end_inset vector spherical harmonics \begin_inset Formula \begin{equation} \iint\vshD{\tau'}{l'}{m'}\left(\uvec r\right)\cdot\vsh{\tau}lm\left(\uvec r\right)\,\ud\Omega=\delta_{\tau'\tau}\delta_{l'l}\delta_{m'm}\label{eq:dual vsh} \end{equation} \end_inset (complex conjugation not implied in the dot product here). In our convention, we have \begin_inset Formula \[ \vshD{\tau}lm\left(\uvec r\right)=\left(\vsh{\tau}lm\left(\uvec r\right)\right)^{*}=\left(-1\right)^{m}\vsh{\tau}{l-}m\left(\uvec r\right). \] \end_inset \end_layout \begin_layout Subsection Cross-sections (single-particle) \end_layout \begin_layout Standard With the \begin_inset Formula $T$ \end_inset -matrix and expansion coefficients of plane waves in hand, we can state the expressions for cross-sections of a single scatterer. Assuming a non-lossy background medium, extinction, scattering and absorption cross sections of a single scatterer irradiated by a plane wave propagating in direction \begin_inset Formula $\uvec k$ \end_inset and (complex) amplitude \begin_inset Formula $\vect E_{0}$ \end_inset are \begin_inset CommandInset citation LatexCommand cite after "sect. 7.8.2" key "kristensson_scattering_2016" literal "true" \end_inset \begin_inset Formula \begin{eqnarray} \sigma_{\mathrm{ext}}\left(\uvec k\right) & = & -\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\Re\left(\rcoeffp{}^{\dagger}\outcoeffp{}\right)=-\frac{1}{2k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\rcoeffp{}^{\dagger}\left(\Tp{}+\Tp{}^{\dagger}\right)\rcoeffp{},\label{eq:extincion CS single}\\ \sigma_{\mathrm{scat}}\left(\uvec k\right) & = & \frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\left\Vert \outcoeffp{}\right\Vert ^{2}=\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\rcoeffp{}^{\dagger}\left(\Tp{}^{\dagger}\Tp{}\right)\rcoeffp{},\label{eq:scattering CS single}\\ \sigma_{\mathrm{abs}}\left(\uvec k\right) & = & \sigma_{\mathrm{ext}}\left(\uvec k\right)-\sigma_{\mathrm{scat}}\left(\uvec k\right)=-\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\left(\Re\left(\rcoeffp{}^{\dagger}\outcoeffp{}\right)+\left\Vert \outcoeffp{}\right\Vert ^{2}\right)\nonumber \\ & & =\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\rcoeffp{}^{\dagger}\left(\Tp{}^{\dagger}\Tp{}+\frac{\Tp{}^{\dagger}+\Tp{}}{2}\right)\rcoeffp{},\label{eq:absorption CS single} \end{eqnarray} \end_inset where \begin_inset Formula $\rcoeffp{}=\rcoeffp{}\left(\vect k,\vect E_{0}\right)$ \end_inset is the vector of plane wave expansion coefficients as in \begin_inset CommandInset ref LatexCommand eqref reference "eq:plane wave expansion" \end_inset . \end_layout \begin_layout Subsection Multiple scattering \end_layout \begin_layout Subsubsection Translation operator \end_layout \begin_layout Subsubsection Numerical complexity, comparison to other methods \end_layout \end_body \end_document