text:
\section{Introduction}

Nano optics deals with light confinement at the nanoscale~\cite{novotny:06,hohenester:20}.  This is achieved by binding light to surface resonances of nanoparticles, such as surface plasmon polaritons for metallic nanoparticles~\cite{maier:07} or surface phonon polaritons for dielectric nanoparticles~\cite{kliewer:74,caldwell:15}.  These resonances come along with strongly localized fields and allow squeezing light into extreme sub-wavelength volumes, which can be exploited for various applications~\cite{barbillon:19}.

Because of the diffraction limit of light, the strongly localized fields cannot be directly imaged in optical microscopy.  In recent years, electron energy loss spectroscopy (\textsc{eels}) has become a highly successful technique for imaging electromagnetic fields at the nanoscale and with high energy resolution~\cite{nelayah:07,kociak:14,colliex:16,polman:19}.  In \textsc{eels} swift electrons pass by or through a nanoparticle and loose with a certain probability energy by exciting surface resonances.  By raster-scanning the electron beam over the specimen and measuring the number of electrons that have lost a certain amount of energy, one obtains information about the electromagnetic fields at the nanoscale~\cite{garcia:10,hohenester:20}.  However, the technique does not provide direct information about the three-dimensional fields but only about the averaged interaction along the entire electron trajectory.

\textsc{eels} tomography is a variant of electron tomography~\cite{midgley:09}, where the three-dimensional structure of a specimen is reconstructed from a collection of transmission electron micrographs for various tilt angles.  In \textsc{eels} the reconstruction is complicated by the fact that the loss does not occur at a specific position of the specimen, but is a highly nonlocal process~\cite{garcia:10}.  \textsc{eels} tomography of surface plasmons was first suggested independently in~\cite{hoerl.prl:13} and \cite{nicoletti:13}, where the latter paper demonstrated experimentally the reconstruction of localized surface plasmon modes for a silver nanocube.  While these seminal papers employed the quasistatic approximation~\cite{garcia:10,hohenester:20}, successive work showed how to extend the scheme to full retardation ~\cite{hoerl:15}, and demonstrated its applicability for single and coupled silver nanoparticles~\cite{hoerl:17,haberfehlner:17}.

In a recent paper~\cite{li:21}, we have brought \textsc{eels} tomography from the optical to the mid-infrared regime, and have demonstrated experimentally the reconstruction of localized surface phonon polaritons for a MgO nanocube.  Contrary to surface plasmon polaritons, the use of the quasistatic approximation is perfectly justified for surface phonon polaritons sustained by nanoparticles with dimensions of a few hundred nanometers.  This considerably simplifies the methodology for the tomographic reconstruction.  While going full circle from the quasistatic tomography of surface plasmon polaritons in our initial work~\cite{hoerl.prl:13} to quasistatic tomography of surface phonon polaritons~\cite{li:21}, we have gained quite some understanding of the critical elements in \textsc{eels} tomography, and our approach has matured considerably.  Time is ripe for a critical re-examination and re-interpretation of our tomography scheme.

In this paper we present a theoretical study of \textsc{eels} tomography for prototypical dielectric nanoparticles.  We submit a tilt series of simulated \textsc{eels} maps to our tomography scheme, in order to extract parameters characterizing the nanophotonic environment.  For this parametrized photonic environment we compute the photonic local density of states (\textsc{ldos}) \cite{garcia:08,novotny:06,hohenester:20}, which is compared with independent simulation results.  From this comparison we examine the strengths and weaknesses of our tomographic reconstruction scheme.

The photonic \textsc{ldos} is a concept borrowed from solid state physics, and accounts for the number of photonic modes per unit frequency and volume.  In free space the photonic \textsc{ldos} is~\cite{novotny:06,hohenester:20} (we use SI units throughout)
\begin{equation}\label{eq:ldos}
  \rho_0(\omega)=\frac{\omega^2}{\pi^2c^3}\,,
\end{equation}
where $\omega$ is the angular frequency and $c$ the speed of light.  The photonic \textsc{ldos} governs the power dissipated by an oscillating dipole through
\begin{equation}
  P_0=\frac{\omega^2p^2}{12\varepsilon_0}\rho_0(\omega)\,,
\end{equation}
where $p$ is the oscillator's dipole moment and $\varepsilon_0$ the free-space permittivity.  Alternatively, we can relate via $P_0=\hbar\omega\,\gamma_0$ the power dissipation to the decay rate $\gamma_0$ of a quantum emitter.  The concept of the photonic \textsc{ldos} comes to full glory in nanophotonics, where the light-matter interaction becomes dramatically enhanced through surface excitations of nanoparticles, such as surface plasmon or phonon polaritons.  The enhancement of the photonic \textsc{ldos} $\rho(\omega)$ can be in the range of hundreds to thousands in comparison to its free-space value $\rho_0(\omega)$~\cite{schuller:10}.  Correspondingly, quantum emitters can transfer energy to the nanophotonic environment more efficiently, and their decay rate or power dissipation $P$ is increased by the \textsc{ldos} enhancement according to
\begin{equation}\label{eq:ldosenhace}
  P:P_0=\rho:\rho_0\,.
\end{equation}
Below we will compute the \textsc{ldos} enhancement $\rho:\rho_0$ using the photonic environment reconstructed from \textsc{eels} maps.  It is obvious that electrons and oscillating dipoles couple quite differently to the nanophotonic environment.  For this reason, the \textsc{ldos} reconstruction from \textsc{eels} data is quite delicate and provides a stringent testbed for our tomography approach.

We have organized our paper as follows.  In Sec.~\ref{sec:theory} we present the theory and methodology of our tomographic reconstruction.  We have tried to keep the presentation as compact and brief as possible, and refer to the literature for the detailed derivations whenever possible.  Some technical issues are transfered to an appendix.  In Sec.~\ref{sec:results} we present the tomography results for smooth and rough nanorods, and compare the reconstructed and the simulated photonic \textsc{ldos}.  Finally, in Sec.~\ref{sec:discussion} we put our tomography into a broader context, examine critically the strengths and weaknesses of our approach, and identify lines for future research.

\section{Theory}\label{sec:theory}

For MgO nanoparticles the surface phonon polariton energies $h\nu$ are of the order of 100 meV, corresponding to a free-space wavelength $\lambda=\nicefrac c\nu\sim 12\,\mu$m.  For nanoparticle dimensions of approximately hundred nanometers we can thus safely introduce the quasistatic approximation~\cite{hohenester:20}, where the electric field is expressed in terms of a quasistatic potential $V(\bm r)$ through $\bm E(\bm r)=-\nabla V(\bm r)$ and we keep the frequency dependence of the permittivity functions $\varepsilon(\omega)$.

\subsection{Green's functions}

\begin{figure}[t]
\includegraphics[width=\columnwidth]{figure01}
\caption{(a) Schematics of Green's function.  In free space the Green's function $G_0(\bm r,\bm r')$ gives the potential at position $\bm r$ for a unit charge located at position $\bm r'$.  In presence of a nanoparticle one must additionally add a reflected Green's function that accounts for the nanoparticle response.  (b) The reflected Green's function can be expanded using a complete set of eigenpotentials $V_k(\bm r)$.  In our tomography scheme we can also start from the modes associated with a simpler reference boundary $\partial\Omega_0$ rather than the actual nanoparticle boundary $\partial\Omega$, and expand the reflected Green's function using the reference modes.  For details see text.}\label{fig:green}
\end{figure}

In the following we consider the problem depicted in Fig.~\ref{fig:green}(a), where a charge located at position $\bm r'$ interacts with a dielectric nanoparticle situated in a background medium with dielectric constant $\varepsilon_0$.  Green's functions provide an elegant and efficient method for solving such problems.  We first introduce the Green's function defined through~\cite{jackson:99,hohenester:20}
\begin{equation}\label{eq:green1}
  \nabla^2 G(\bm r,\bm r')=-\delta(\bm r-\bm r')\,,
\end{equation}
which gives the potential at position $\bm r$ for a unit charge located at position $\bm r'$.  In an unbounded medium the Green's function would be given by the usual expression
\begin{equation}\label{eq:green2}
  G_0(\bm r,\bm r')=\frac 1{4\pi|\bm r-\bm r'|}\,,
\end{equation}
and the potential associated with a charge distribution $\rho(\bm r)$ can be expressed as
\begin{equation}\label{eq:vinc}
  V_{\rm inc}(\bm r)=\int \frac{\rho(\bm r')}{4\pi\varepsilon_0|\bm r-\bm r'|}\,d^3r'\,.
\end{equation}
In presence of the nanoparticle this incoming potential will induce a reflected potential associated with the particle response.  To account for this, we split the total Green's function into two parts
\begin{equation}\label{eq:green3}
  G(\bm r,\bm r')=G_0(\bm r,\bm r')+G_{\rm refl}(\bm r,\bm r')\,,
\end{equation}
where the reflected part is a solution of Laplace's equation which is chosen such that Maxwell's boundary conditions are fulfilled at the nanoparticle boundary.  Suppose for a moment that the reflected Green's function is at hand.  It can then be shown that in \textsc{eels} the loss probability is related to the reflected Green's function via~\cite{garcia:10,hohenester:20}
\begin{equation}\label{eq:eels1}
  \Gamma(\bm R_0,\omega)=-\frac 1{\pi\hbar}\int \mbox{Im}\Big[\rho_{\rm el}^*(\bm r)
  G_{\rm refl}(\bm r,\bm r')\rho_{\rm el}(\bm r')\Big]\,d^3rd^3r'\,,
\end{equation}
where $\bm R_0=(x_0,y_0)$ is the impact parameter of the electron beam propagating along the $z$ direction (aloof geometry), $\hbar\omega$ is the loss energy and $\rho_{\rm el}(\bm r)$ the charge distribution of the swift electron.  The term in brackets of Eq.~\eqref{eq:eels1} accounts for a self-interaction process where the swift electron polarizes the nanoparticle, and the polarization acts back on the electron.  This nonlocal response is mediated by the reflected Green's function.  Similarly, the power dissipated by a dipole oscillating with frequency $\omega$ becomes~\cite{hohenester:20}
\begin{equation}\label{eq:ldos1}
  P=P_0-\frac\omega{2\varepsilon_0} \mbox{Im}\Big[
  (\bm p\cdot\nabla)(\bm p\cdot\nabla')G_{\rm refl}(\bm r,\bm r')\Big]_{\bm r=\bm r'=\bm r_0}\,,
\end{equation}
where $P_0$ is the free-space dissipation, $\bm p$ is the dipole moment, and $\bm r_0$ the position of the dipole.  The ratio $P:P_0$ gives the enhancement of the photonic \textsc{ldos}, see also Eq.~\eqref{eq:ldosenhace}.  The expressions given in Eqs.~\eqref{eq:eels1} and \eqref{eq:ldos1} are two examples for the enhancement of light-matter interactions in presence of nanoparticles, and show that the nanophotonic environment is fully characterized upon knowledge of the reflected Green's function.

\subsection{Eigenmode decomposition}

A powerful and convenient representation of the reflected Green's function is in terms of geometric eigenmodes $u_k(\bm s)$ and eigenvalues $\lambda_k$, where $\bm s$ is a position located on the boundary of the nanoparticle~\cite{ouyang:89,boudarham:12,hohenester:20}.  These eigenmodes form a complete set of basis functions.  To each eigenmode we can associate an eigenpotential
\begin{equation}\label{eq:eigpot}
  V_k(\bm r)=\oint_{\partial\Omega}\frac{u_k(\bm s')}{4\pi|\bm r-\bm s'|}\,dS'\,,
\end{equation}
which is a solution of Laplace's equation that fulfills Maxwell's boundary conditions at the nanoparticle boundary.  We can then decompose the reflected Green's function outside the nanoparticle in terms of these eigenpotentials via~\cite{boudarham:12,hohenester:20}
\begin{equation}\label{eq:grefl1}
  G_{\rm refl}(\bm r,\bm r')=-\sum_k V_k(\bm r)
  \left[\frac{\lambda_k+\frac 12}{\Lambda(\omega)+\lambda_k}\right]V_k(\bm r')\,,
\end{equation}
where $\Lambda(\omega)$ is an expression that solely depends on the permittivities of the nanoparticle and the embedding medium.  Inserting Eq.~\eqref{eq:grefl1} into the \textsc{eels} loss probability of Eq.~\eqref{eq:eels1} leads us to
\begin{equation}\label{eq:eels2}
  \Gamma(\bm R_0,\omega)=\frac 1{\pi\hbar\varepsilon_0}\sum_k
  L_k(\omega)\left|\int \rho_{\rm el}(\bm r)V_k(\bm r)\,d^3r\right|^2\,,
\end{equation}
with the lineshape function
\begin{equation}
  L_k(\omega)=\mbox{Im}\left[\frac{\lambda_k+\frac 12}{\Lambda(\omega)+\lambda_k}\right]\,.
\end{equation}
Eq.~\eqref{eq:eels2} is a particularly useful decomposition of the loss probability in terms of surface phonon polariton eigenmodes.  Each eigenmode contributes with the lineshape function $L_k(\omega)$ and the oscillator strength given by the square modulus term, which is governed by the interaction energy between the charge distribution of the swift electron and the eigenpotential $V_k(\bm r)$.  Similarly, the power dissipated by an oscillating dipole of Eq.~\eqref{eq:ldos1} can be decomposed into eigenmodes via
\begin{equation}\label{eq:ldos2}
  P=P_0+\frac{\omega}{2\varepsilon_0}\sum_k L_k(\omega)\big|\bm p\cdot\nabla V_k(\bm r)\big|^2_{\bm r=\bm r_0}\,,
\end{equation}
with a corresponding interpretation in terms of lineshape functions and oscillator strengths.  From the dissipated power one can obtain the photonic \textsc{ldos} using Eqs.~\eqref{eq:ldos} and \eqref{eq:ldosenhace}, where one often additionally averages over all dipole orientations to account for the random orientation of quantum emitters in typical experiments~\cite{novotny:06}.

\subsection{Tomographic reconstruction of eigenmodes}

It is apparent from Eqs.~\eqref{eq:eels2} and \eqref{eq:ldos2} that we can compute the \textsc{eels} loss probability $\Gamma(\bm R_0,\omega)$ and the \textsc{ldos} enhancement $P:P_0$, or any other related response function, once the geometric eigenmodes $u_k(\bm s)$ and the lineshape function $L_k(\omega)$ are at hand.  Expressed differently, the nanophotonic environment is fully characterized upon knowledge of $u_k(\bm s)$ and $L_k(\omega)$.  We can now formulate the goal of our tomography approach.  Suppose that we are in possession of the \textsc{eels} loss probabilities $\Gamma(\bm R_0,\omega)$, ideally for various impact parameters and electron propagation directions, but don't know the eigenmodes $u_k(\bm s)$ and lineshape functions $L_k(\omega)$: can we obtain through solution of an inverse problem a viable approximation for $u_k(\bm s)$ and $L_k(\omega)$?  And if yes, how?

\begin{figure*}[t]
\centerline{\includegraphics[width=1.6\columnwidth]{figure02}}
\caption{Schematics of tomographic reconstruction for a rough nanorod.  The reference boundary is formed by a smooth rod, see panels on top of the figure.  The experimental \textsc{eels} maps $\Gamma_{\rm exp}$ are obtained for a specific loss energy and for various rotation angles, we only keep aloof electron trajectories that do not penetrate the smooth rod.  We start with some initial guess for the optimization parameters $L_k$, $\mathbb{Q}$ and compute the reprojected \textsc{eels} maps $\Gamma$ using Eq.~\eqref{eq:eels3}.  These parameters are optimized until a local minimum is reached by the optimization algorithm.  In the lowest row we show the relative error $|\Gamma_{\rm exp}-\Gamma|:\Gamma_{\rm exp}$ between the experimental and optimized maps.  The solid lines indicate the contours for an error of~0.1\%.  Once the parameters $L_k$, $\mathbb{Q}$ are at hand, we can compute other quantities such as the photonics \textsc{ldos}.}\label{fig:cost}
\end{figure*}

\subsubsection{Optimization for modes on nanoparticle boundary}

Consider first the situation that the nanoparticle boundary is known and that we are seeking for the linshape functions and eigenmodes $L_k$, $u_k(\bm s)$.  This corresponds to the situation previously investigated in~\cite{li:21}.  Let $u_\ell^0(\bm s)$ be a complete set of basis functions on the boundary.  We shall refer to these modes as \textit{reference modes}.  As shown in Appendix~\ref{sec:ortho}, the eigenpotentials of Eq.~\eqref{eq:eigpot} can be expanded in terms of these modes via
\begin{equation}\label{eq:eigpotref1}
  V_k(\bm r)=\sum_\ell\mathbb{Q}_{k\ell}\oint_{\partial\Omega}
  \frac{ u^0_\ell(\bm s')}{4\pi|\bm r-\bm s'|}\,dS'\,,
\end{equation}
with $\mathbb{Q}$ being an orthogonal matrix.  We can now formulate the tomographic reconstruction scheme for a given set of experimental \textsc{eels} maps.

\begin{enumerate}

\item Find some reference modes $u_\ell^0(\bm s)$ whose gross features are expected to be similar to those of the true eigenmodes $u_k(\bm s)$.  This point is irrelevant for a complete basis, but becomes crucial for actual reconstructions where the basis has to be truncated.

\item Start with some initial guess for the lineshape function $L_k$ and orthogonal matrix $\mathbb{Q}$, and compute the reprojected maps via Eq.~\eqref{eq:eels2}.  Use an optimization routine for $L_k$, $\mathbb{Q}$ to obtain the best possible agreement between experiment and reprojection.  Note that in principle $L_k(\omega)$ depends on frequency, but for a fixed loss energy the lineshape functions can be treated as mere numbers.

\item Use the optimized parameters to compute other quantities, such as the photonic \textsc{ldos}.
\end{enumerate}

\subsubsection{Optimization for modes on reference boundary}

The above scheme can be also generalized to cases where the true nanoparticle boundary $\partial\Omega$ is not known or is too complicated to be used in actual reconstructions.  We start by introducing a reference boundary $\partial\Omega_0$ that fully encapsulates the nanoparticle, see also Fig.~\ref{fig:green}(b). In our modified approach we are not aiming for a reconstruction of the eigenmodes $u_k(\bm s)$ themselves, but of the eigenpotentials of Eq.~\eqref{eq:eigpot} outside of the reference boundary.  There they can be expressed as generic solutions of Laplace's equation~\cite{jackson:99}
\begin{equation}\label{eq:eigpotref2}
  V_k(\bm r)=\oint_{\partial\Omega_0}\frac{\sigma_k(\bm s')}{4\pi|\bm r-\bm s'|}\,dS'\,,
\end{equation}
where $\sigma_k(\bm s)$ specifies the normal derivative of the potential on $\partial\Omega_0$ (von Neumann boundary condition).  We can now use a complete set of basis functions $u_\ell^0(\bm s)$ on $\partial\Omega_0$ for the expansion of $\sigma_k(\bm s)$ to arrive at
\begin{equation}\label{eq:eigpotref3}
  V_k(\bm r)=\sum_\ell\mathbb{Q}_{k\ell}\oint_{\partial\Omega_0}
  \frac{ u^0_\ell(\bm s')}{4\pi|\bm r-\bm s'|}\,dS'\,,
\end{equation}
where $\mathbb{Q}$ is a non-orthogonal matrix formed by the expansion coefficients.  The tomographic reconstruction can now be performed in complete analogy to the scheme presented above, with the only exception that $\mathbb{Q}$ has to be replaced by a non-orthogonal matrix.

\begin{figure*}[t]
\centerline{\includegraphics[width=2\columnwidth]{modes}}
\caption{Reference and reconstructed modes.  In our tomographic reconstruction we use as reference modes $u_\ell^0(\bm s)$  the eigenmodes of the Laplace-Beltrami operator.  Using the optimized parameters $L_k$, $\mathbb{Q}$ we mix the modes to obtain the reconstructed modes shown on the right hand side for the dipole and quadrupole resonances.  For the smooth rod, $\mathbb{Q}$ is an orthogonal matrix of size $n\times n$, where $n$ is the truncation number of the basis.  For the rough rod, $\mathbb{Q}$ is a full matrix of size $m\times n$, where $m$ is the number of eigenpotentials to be reconstructed.  From the knowledge of $\mathbb{Q}$ we can compute the geometric eigenpotentials $V_k(\bm r)$ outside the reference boundary.  The bar plot on the right hand side reports the reconstructed lineshape parameters for the dipole (blue) and quadrupole (red) resonances, the modes are sorted in decreasing order of $L_k$ and the largest contributions are due to the modes shown in the insets.}\label{fig:modes}
\end{figure*}

\subsubsection{Optimization loop}

In the following we discuss the optimization procedure in slightly more detail, see also Figs.~\ref{fig:cost} and \ref{fig:modes}.  We provide a unified description for the optimizations using modes defined on either the nanoparticle or reference boundary.  In the first case, $\mathbb{Q}$ is an orthogonal matrix.  In our computational approach we have to truncate the basis and keep only $n$ representative modes, where $n$ is of the order of several tens to hundreds.  Correspondingly, $\mathbb{Q}_{n\times n}$ is a matrix of size $n\times n$, see also Appendix~\ref{sec:ortho} for the parametrization of this matrix.  In the case of a reference boundary, $\mathbb{Q}$ is a full matrix.  In principle we can now use different truncation numbers $m$ and $n$ for the reconstructed eigenpotentials and basis functions, respectively, and $\mathbb{Q}_{m\times n}$ becomes a matrix of size $m\times n$.  In most cases it is sufficient to consider around ten eigenpotentials, whereas the truncation number for the basis should be chosen considerably larger.  Let
\begin{equation}
  x_i=\big\{\bm R_0^{(i)},\theta^{(i)}\big\}
\end{equation}
be a set of impact parameters and tilt angles for a fixed loss energy, and $\Gamma_{\rm exp}(x_i)$ the corresponding experimental \textsc{eels} maps.  We only consider aloof electron trajectories that do not penetrate the nanoparticle.  The interaction energy between the swift electron and a reference mode $u_\ell^0(\bm s)$ is
\begin{equation}
  \mathscr{V}_\ell(x_i)=\int\rho_{\rm el}(\bm r)V_\ell^0(\bm r)\,d^3r=
   \oint_{\partial\Omega_0}V_{\rm el}(\bm s)u_\ell^0(\bm s)\,dS\,,
\end{equation}
where $V_{\rm el}(\bm r)$ is the potential associated with the charge distribution $\rho_{\rm el}(\bm r)$.  When the nanoparticle boundary is known, the reference boundary in the above boundary integral is identical to $\partial\Omega$.  The loss probability of Eq.~\eqref{eq:eels2} can then be written in the compact form
\begin{equation}\label{eq:eels3}
  \Gamma(x_i;L_k,\mathbb{Q})=\frac 1{\pi\hbar\varepsilon_0}\sum_k\sum_{\ell,\ell'}
  \big(\mathbb{Q}_{k\ell}\mathscr{V}_\ell(x_i)\big)L_k
  \big(\mathbb{Q}_{k\ell'}\mathscr{V}_{\ell'}(x_i)\big)\,.
\end{equation}
We can now define a cost function
\begin{equation}\label{eq:cost}
  J(L_k,\mathbb{Q})=\frac 12\sum_i\Big|\Gamma_{\rm exp}(x_i)-\Gamma(x_i;L_k,\mathbb{Q})\Big|^2
  \longrightarrow\mbox{min}\,,
\end{equation}
that gives the ``distance'' between the experimental and reprojected \textsc{eels} maps.  This cost function is  submitted to an optimization routine, such as a conjugate-gradient or quasi-Newton one~\cite{press:02}, which provides us with the optimized expressions for $L_k$, $\mathbb{Q}$.  Some details about the parametrization of the orthogonal matrix, as well as the computation of the derivative of the cost function with respect to the optimization parameters are given in Appendix~\ref{sec:ortho}.

\section{Results}\label{sec:results}

In~\cite{li:21} we have applied our tomography scheme to experimental \textsc{eels} maps for a MgO nanocube.  In this work we proceed differently and investigate the working principle of our tomography scheme using simulated data only.

\begin{enumerate}

\item We first compute for each loss energy \textsc{eels} maps for a series of rotation angles, see also Fig.~\ref{fig:cost}.  To be consistent with our previous notation, we denote these simulated \textsc{eels} maps as $\Gamma_{\rm exp}$, and will refer to them as \textit{experimental \textsc{eels} maps}.

\item These maps are submitted to our tomography scheme based on Eq.~\eqref{eq:cost}, in order to obtain the optimized parameters $L_k$, $\mathbb{Q}$ that specify the nanophotonic environment.

\item Using Eq.~\eqref{eq:ldos2} together with the optimized parameters, we compute the photonic \textsc{ldos}, and will refer to it as the \textit{reconstructed photonic} \textsc{ldos}.

\item Using Eq.~\eqref{eq:ldos1} we compute the photonic \textsc{ldos} directly, with a simulation approach to be discussed below, and will refer to it as the \textit{simulated photonic} \textsc{ldos}.
\end{enumerate}

\begin{figure}[b]
\centerline{\includegraphics[width=\columnwidth]{spectra}}
\caption{Loss spectra for smooth and rough nanorod, and for impact parameters located on the long (blue) and short (red) rod axis, see inset.  We consider aloof electron trajectories with a propagation direction out of the image plane.  One observes a dipole resonance around 70 meV, a quadrupole resonance around 80 meV, and a peak attributed to a multitude of modes around 90 meV.}\label{fig:spectra}
\end{figure}

\noindent For ideal reconstruction the simulated and reconstructed \textsc{ldos} maps should be identical.  Any deviation between the two maps can thus be attributed to deficiencies of our approach, caused for instance by the truncation of the reference basis $u_\ell^0(\bm s)$ or a trapping of the optimization algorithm in a local minimum.

We apply our tomography scheme to prototypical systems of a smooth and rough nanorod with a diameter to length ratio of approximately $1:2.5$, see also Fig.~\ref{fig:spectra} and \cite{lourenco:17} for a detailed discussion of the rod modes.  The rough rod has been by generated by adding stochastic height variations to the smooth surface of an ideal nanoparticle, following the prescription given in~\cite{truegler.prb:11}.  We shall not be concerned whether such nanoparticles can indeed be fabricated with the material system under investigation.  As we are working within the quasistatic regime, the actual size of the nanorods is irrelevant and the results can be easily scaled to any size.

\subsection{Computational details}

All our simulations are performed with the quasistatic classes of the \textsc{nanobem} toolbox~\cite{hohenester.cpc:22}, which is based on a Galerkin scheme with linear shape elements.  See e.g.~\cite{hohenester:20} for a detailed discussion.  The parametrization of the MgO dielectric function is the same as in~\cite{hohenester.prb:18,li:21}.  The nanorod boundaries are discretized using more than 3000 boundary elements of triangular shape.  We checked that for such fine discretizations we obtained converged results.  As for the \textsc{eels} simulations, we consider the limit of large electron velocities $v$ where the potential for a swift electron with impact parameter $\bm R_0$ takes the form
\begin{equation}
  V_{\rm el}(\bm r;\bm R_0)=-\frac{e}{2\pi v}\ln\big|\bm R-\bm R_0\big|\,,
\end{equation}
with $e$ being the elementary charge and $\bm R=(x,y)$.  We have previously shown in Fig.~S9 of \cite{li:21} that this simplified expression gives almost the same results as simulations based on the full Maxwell's equations.

As for the reference modes $u_\ell^0(\bm s)$, we did not choose the usual geometric eigenmodes~\cite{boudarham:12,hohenester:20} for two reasons.  First, in order to demonstrate that our approach indeed works for any meaningful set of basis functions.  Second, we observed that the geometric eigenmodes computed with the \textsc{nanobem} toolbox are often strongly localized around sharp corners or edges, such that a large number of such modes would be needed for a useful expansion.  In this work, we choose for $u_\ell^0(\bm s)$ the eigenmodes of the Laplace-Beltrami operator, which is a generalization of the Laplace operator for curved boundaries and is known to provide extremely smooth basis functions~\cite{bobenko:07}.  The modes were additionally orthogonalized using Eq.~\eqref{eq:orthoref}.

In our optimization approach we truncate the Laplace-Beltrami basis using the $n$ modes of highest eigenvalue, where a value of $n\approx 100$ turned out to be a good compromise between reasonably fast optimizations and sufficiently accurate results.  The optimization was performed with the builtin \textsc{matlab} function \texttt{fminunc} using a quasi-Newton algorithm together with a relatively small function and optimality tolerance of $10^{-8}$.  In all our simulation we typically needed about 2000 iterations to reach a local minimum.

\subsection{Smooth rod}

We start by discussing the smooth rod shown in Fig.~\ref{fig:spectra}.  The loss spectra exhibit three peaks, which can be attributed to a dipolar mode (70 meV), a quadrupolar mode (80 meV), and a peak that is composed of a multitude of modes (88 meV).  For the smooth rod, the reference boundary $\partial\Omega_0$ is identical to the true nanoparticle boundary $\partial\Omega$.  Note that the Laplace-Beltrami eigenmodes provide a (truncated) basis that does not coincide with the true geometric eigenmodes.  Simulated and reprojected maps originating from our optimization algorithm are typically extremely similar, see lowest row in Fig.~\ref{fig:cost} for the more difficult case of the rough nanorod.

\begin{figure}[t]
\centerline{\includegraphics[width=\columnwidth]{ldos01}}
\caption{Simulated and reconstructed \textsc{ldos} maps for the different loss energies reported in the panels and in the different planes indicated on top of the figure.  The electron propagation direction is out of the image plane and the lines at the rod centers indicate the tilt axis.  (a) Simulated \textsc{ldos} maps for dipole mode, (b,c) reconstructed \textsc{ldos} maps for different numbers $n$ of Laplace-Beltrami eigenmodes.  Same for (d,e) quadrupole resonance and (f,g) multitude of modes.  The \textsc{ldos} maps in the first column are displayed for a logarithmic color scale, in the other columns we use a linear color scale.  All maps are scaled to the maxima of the simulated maps.  The solid lines report the contours for 20\% of the maximum of the simulated \textsc{ldos} in the respective planes.}\label{fig:ldos1}
\end{figure}

\begin{figure}[t]
\centerline{\includegraphics[width=\columnwidth]{ldos02}}
\caption{Cuts through the \textsc{ldos} maps shown in Figs.~\ref{fig:ldos1}(a,b) along the long rod axis at $y=0$.  The solid lines report simulation results, the dashed lines show the reconstructed results.  The \textsc{ldos} enhancements are given in arbitrary units, with a constant prefactor for the reconstructed \textsc{ldos} maps.  For a discussion see text.  Larger \textsc{ldos} enhancements correspond to positions closer to the nanorod, the colors are in agreement with those of the planes shown on top of Fig.~\ref{fig:ldos1}.  Distances are given in units of the rod length $L$.}\label{fig:ldos2}
\end{figure}

Figure~\ref{fig:ldos1} shows the simulated and reconstructed \textsc{ldos} maps in the symmetry plane (left column) and in planes away from the rod (other columns), and for the loss energies reported in the figure.  We first consider the dipole mode shown in panel~(a).  The \textsc{ldos} can be interpreted for an oscillating dipole as the enhancement of the dissipated power, see Eq.~\eqref{eq:ldos1}, throughout we average over all possible dipole orientations.  Close to the rod, an oscillating dipole couples with comparable strength to all surface phonon polariton modes.  This can be seen both in the symmetry plane of the rod (first column, logarithmic color scale), as well as in the plane closest to the rod (second column, linear color scale), where the photonic \textsc{ldos} is large and unstructured close to the rod boundary.  When moving away from the rod (other columns from left to right), the coupling strength between the oscillating dipole and the rod resonance modes have different distance dependencies, which are governed by the oscillator strengths given in Eq.~\eqref{eq:ldos2}.  For the chosen loss energy the dipolar rod mode becomes strongest at larger distances, as can be inferred from the two lobes in the \textsc{ldos} maps located at the rod caps.

\begin{figure*}[t]
\centerline{\includegraphics[width=2\columnwidth]{ldos03}}
\caption{Same as Fig.~\ref{fig:ldos1} but for a rough nanorod and for the (a--c) dipolar and (d--f) quadrupolar rod resonance.  In the recon\-struction we consider $n=200$ reference modes for a smooth nanord (black contour shown on top) and $m=20$ modes to be reconstructed.  We compare \textsc{ldos} values in planes (a--f) parallel and (a*--f*) perpendicular to the electron propagation direction.  The lines and dots in the rod center indicate the tilt axis.  In panels (b,e) we conisder a tilt series where the nanoparticle is rotated around the $y$-axis only, whereas in panels (c,f) we additionally consider a rotation around the $x$-axis by 90$^\circ$ followed by the same tilt series around $y$.}\label{fig:ldos3}
\end{figure*}

Figures~\ref{fig:ldos1}(b,c) show results for the reconstructed \textsc{ldos} using (b) $n=100$ and (c) $n=20$ Laplace-Beltrami reference modes.  Further away from the rod, the simulated and reconstructed results agree well for both truncation numbers $n$.  For distances closer to the rod, the larger number of eigenmodes provides better agreement.  This is in accordance to our previous reasoning that oscillating dipoles close to the rod couple to a larger number of eigenmodes, and thus a larger number of modes is needed for the reconstruction.

In Fig.~\ref{fig:ldos2} we give a quantitative comparison between the simulated (full lines) and reconstructed (dashed lines) \textsc{ldos} values for cuts along the long rod axis and for dipole positions outside the nanoparticle.  The true \textsc{ldos} enhancement would depend on the actual size of the nanorod, for simplicity we give the results in arbitrary units.  Also the reconstruced \textsc{ldos} cuts are scaled by a constant factor, where it is not obvious how this factor could be obtained in absence of \textsc{eels} loss probabilities given in absolute numbers.  We here do not enter into the question of how to extract the absolute numbers of the reconstructed \textsc{ldos}.  Besides this unknown prefactor, the simulated and reconstructed \textsc{ldos} values agree extremely well, with the possible exception of the smallest distances where a larger number of eigenmodes might be needed.

Finally, in the remaining panels of Fig.~\ref{fig:ldos1} we compare the simulated and reconstructed \textsc{ldos} for the (d,e) quadrupolar rod mode and the (f,g) multitude of modes.  It can be seen that the reconstruction works well for the quadrupolar mode.  Comparison with results for $n=20$ (not shown) reveal that in this case a larger number of eigenmodes is strictly needed to obtain good agreement.  For the multitude of modes shown in panels (f,g) the agreement between simulation and reconstruction is reasonable but not overly good.  In particular for the smallest distances the reconstructed maps show sharp or asymmetric features, which are absent in the simulated maps.  From these results we conclude that the \textsc{ldos} reconstruction works best for loss peaks that are governed by a few modes only.

\begin{figure}[t]
\centerline{\includegraphics[width=\columnwidth]{ldos04}}
\caption{\textsc{ldos} maps for dipole mode of rough nanorod and for different $(m,n)$ cutoffs used in the optimization.  Here $m$ is the number of eigenpotentials to be reconstructed and $n$ is the number of basis modes.  As can be seen, the reconstruced \textsc{ldos} does not depend decisevly on the chosen parameters.}\label{fig:ldos4}
\end{figure}

\subsection{Rough rod}

The case of the rough rod shown in Fig.~\ref{fig:spectra} is considerably more difficult.  We keep considering the \textit{same} reference modes as for the smooth rod, and select the reference boundary $\partial\Omega_0$ such that it fully encapsulated the boundary $\partial\Omega$ of the rough rod.  Note that this reference boundary is identical to the one of the smooth rod.  Fig.~\ref{fig:cost} shows for the dipolar mode the simulated (``experimental'') \textsc{eels} maps and the reprojected ones.  The relative error between these maps is small throughout.

In Fig.~\ref{fig:ldos3} we show the simulated and reconstructed \textsc{ldos} maps for the rough nanorod.  We compare different planes that are (a--f) parallel and (a*--f*) perpendicular to the electron beam direction.  The main difference between these two configurations is that in the parallel case we reconstruct the \textsc{ldos} throughout in regions through which swift electrons have traveled.  In contrast, for the perpendicular case we reconstruct the \textsc{ldos} also in planes above the nanoparticle through which no electron has traveled because of our restriction to aloof trajectories.

Let us consider the parallel case first.  With the possible exception of the smallest distance, the agreement between simulated and reconstructed \textsc{ldos} maps is extremely good, both for the dipolar and quadrupolar mode.  Both asymmetries as well as hot spots, caused by localized fields in the vicinity of protrusions of the rough rod, are well reproduced by our tomography scheme.  Things somewhat change for the perpendicular geometry shown in panels (b*) and (e*), where the comparison is reasonable but not overly good.  We performed additional simulations where the tilt series for $\Gamma_{\rm exp}$ is complemented by \textsc{eels} maps where the nanorod is first rotated around the $x$-axis by $90^\circ$ before being submitted to the same tilt series around $y$.  As can be seen in panels (c*) and (f*), with this procedure we again obtain extremely good agreement between simulated and reconstructed \textsc{ldos} maps.  This shows that our tomography scheme works best for regions through which electrons have traveled.

We finally investigate in Fig.~\ref{fig:ldos4} the impact of the cutoff parameters $(m,n)$ on the reconstructed \textsc{ldos} for the dipole mode.  Recall that $m$ is the number of eigenpotentials to be reconstructed, and $n$ is the cutoff parameter for the basis functions.  Close to the particle (second column) a larger number $n$ of basis states leads to a better agreement with the simulated \textsc{ldos}, shown in the first row.  When moving away from the nanoparticle, the agreement between simulated and reconstruced \textsc{ldos} maps is very good for all chosen simulation parameters.  This demonstrates that our reconstruction scheme is robust and that the optimization results do not depend decisively on the input parameters.

\section{Discussion}\label{sec:discussion}

In the previous sections we have presented the methodology of our tomographic reconstruction scheme and have investigated the approach for prototypical nanophotonic structures.  In this section we start by discussing our scheme within a broader context, and then address limitations, dos and dont's, as well as extensions of our tomographic reconstructions.

\subsection{Working principle}

\begin{figure}[t]
\centerline{\includegraphics[width=\columnwidth]{basic}}
\caption{Working principle of our tomography scheme.  The approach consists of the triad of experiment, resonance modes, and references modes.  The experimental \textsc{eels} maps for various tilt angles provide the basic resource for the reconstruction of the photonic environment.  The resonance modes are used to formulate the theory underlying the reconstruction, the reference modes provide the parametrization of the photonic environment and are used for the actual reconstruction.  The parameters $L_k$, $\mathbb Q$ are obtained through an optimization procedure in order to minimize the difference between the measured and reprojected maps.  As the potentials outside the nanoparticle are solutions of Laplace's equation, we can employ a boundary element method (\textsc{bem}) scheme to express the potentials through their values on a boundary.}\label{fig:basic}
\end{figure}

The basic working principle of our tomographic reconstruction is shown in Fig.~\ref{fig:basic} and consists of the triad formed by experiment, resonance modes, and reference modes.  In short, the resonance modes are needed to formulate the theory, and the reference modes to provide a parametrization of the nanophotonic environment and to perform the actual reconstruction.  The experimental data are the primary resource for the reconstruction.  For this reason, the quality of the experimental data directly influences the quality of the tomographic reconstruction.  Some further considerations about experiments will be given below.

\subsubsection{Resonance modes}

The nanophotonic environment outside the nanoparticle is fully characterized in terms of the the reflected Green's function of Eq.~\eqref{eq:grefl1}, which we repeat here in compact form
\begin{equation}\label{eq:grefl2}
  G_{\rm refl}(\bm r,\bm r')=-\sum_k V_k(\bm r)\big(M_k+iL_k\big)V_k(\bm r')\,.
\end{equation}
$M_k$ and $L_k$ are the real and imaginary parts of the term given in brackets of Eq.~\eqref{eq:grefl1}.  The eigenpotentials $V_k(\bm r)$ provide the preferred physical basis, only with this basis the reflected Green's function can be written in the diagonal form of Eq.~\eqref{eq:grefl2}.  A similar decomposition of the Green's function can be also obtained in the retarded case when using quasinormal modes~\cite{leung:94,lalanne:19,kristensen:20}, as will be discussed below.  For this reason, from here on we use the more general expression of resonance modes rather than geometric eigenmodes, for which we have developed our theory so far.

With these modes, both the \textsc{eels} loss probability of Eq.~\eqref{eq:eels2} as well as the power dissipation of an oscillating dipole, Eq.~\eqref{eq:ldos2}, can be written as the sum over individual loss channels.  With any other basis one would obtain some kind of mixing between different modes.  This particular form has the additional advantage that the lineshape function $L_k$ is always positive, at least for lossy materials, which can be used in our optimization procedure as a constraint, see Appendix~\ref{sec:ortho}.  Note that our tomography scheme only allows for the reconstruction of $L_k$, which accounts for the loss properties of the nanophotonic environment, but not for the propagation properties described by $M_k$.  As \textsc{eels} and \textsc{ldos} account for energy losses of electrons and oscillating dipoles, respectively, this is not a problem here.  However, additional experimental input or a reconstruction for various loss energies together with a Kramer-Kronig analysis would be needed for a reconstruction of $M_k$.

To summarize this part, resonance modes are needed to formulate the abstract theory, without making contact to the actual shape or composition of the nanoparticle.  Without resonance modes it would be unclear which properties of the nanophotonic environment govern \textsc{eels} and \textsc{ldos}, and which properties can be reconstructed using an inverse scheme.  However, at no point of our approach we require explicit knowledge of the actual form of the resonance modes or lineshape functions.

\subsubsection{Reference modes}

The reference modes are the device needed for the actual reconstruction.  They allow for a suitable paramet\-rization of the nanophotonic environment, where the viable parameters can be extracted from the optimization loop using the experimental and reprojected \textsc{eels} maps.  In principle, for a complete basis the choice of the reference modes is irrelevant.  However, in all practical cases one has to truncate the basis, which should thus be based on an educated guess and should include the gross features of the expected resonance modes from the outset.

\subsubsection{Boundary element method approach}

Although all our calculations presented here and elsewhere~\cite{hoerl.prl:13,hoerl:15,hoerl:17,haberfehlner:17,li:21} have been performed using a boundary element method (\textsc{bem}) approach, it doesn't play an exceptional role in our tomography scheme.  The reference modes are fixed by specifying their values on a properly chosen reference boundary, see Fig.~\ref{fig:basic}.  Away from the boundary the modes propagate according to Laplace's equation, see Eq.~\eqref{eq:eigpot}, or the source-free Maxwell's equations in the retarded case.  This propagation is reminiscent of Huygens' principle for the wavefront propagation in free space and can be well described within \textsc{bem}, but otherwise our tomography makes no particular use of it.

\subsection{Frequently asked questions}

All our cards are on the table now.  Up to here, we have presented and examined our tomography approach in some detail, and have put it into a broader context.  However, a number of open or not fully clear issues remains.  In the following we address these issues in the form of frequently asked questions.  As will become apparent, only some of these questions can be answered definitely while others remain open.  In this sense, the following discussion is meant to summarize our present understanding of the field, to make aware where things can go wrong, and to identify directions for future research.

\textit{How much preknowledge is needed?}  Any tomography or inverse scheme requires some sort of preknowledge, less preknowledge usually makes an approach more general and powerful.  In our case, we assume that the nanophotonic enviromnent can be expressed in terms of resonance modes, and that the potentials away from the boundary propagate as solutions of Laplace's equation.

\textit{How many reference modes (and which) are required?}  For any practical reconstruction one has to truncate the basis.  The proper choice and truncation of the reference basis thus enters as an additional preknowledge.  For spectrally isolated resonances often a few tens of modes suffice, while in other cases up to hundred modes might be needed.  We are not aware of any general approach for determining the correct number of reference modes, so we advice potential users to vary the number and to obtain the best cutoff parameter on a case-to-case basis.

\textit{Are more reference modes always better?}  More modes slow down the optimization and require more iterations until reaching convergence.  With the simulated \textsc{eels} data used in this paper the quality of the reconstruction did not depend decisively on the truncation number.  However, things might change for noisy experimental data because higher modes come along with spatially more localized potential variations, and at some point one might end up reconstructing these noisy features rather than the real physical ones.

\textit{What is the typical computational cost?}  Depending on the truncation number, typical optimization times range from one to several minutes on a normal computer.  The code developments on top of a \textsc{bem} solver, such as the \textsc{nanobem} one~\cite{hohenester.cpc:22}, are moderate.  In the future we consider publishing our code to make it accessible to interested users.

\textit{Are there differences between \textsc{eels} and \textsc{ldos}?}  The question is somewhat odd, obviously \textsc{eels} accounts for the energy loss of swift electrons and \textsc{ldos} accounts for the enhancement of the decay rate of oscillating dipoles.  On the other hand, the interaction potential of the swift electron to the nanoparticle has a $\log(r)$ spatial dependence while the dipole has a $\nicefrac 1{r^2}$ dependence.  For this reason, \textsc{eels} maps are governed by the long-range features of the potential and \textsc{ldos} maps by the short-range features.  The prediction of \textsc{ldos} maps from experimental \textsc{eels} maps is thus a challenging and difficult task, and the good agreement between simulated and reprojected \textsc{ldos} maps reported in this work should not be taken as granted.

\textit{Does the optimization always succeed?}  In all reconstructions considered in this work the optimization algorithm ended up in a minimum.  However, there is no guarantee that this is a global minimum.  Our results never depended decisively on the initial values for $L_k$, $\mathbb{Q}$, which we initially set equal to one.  Note that zeros would be a bad choice because the derivatives of the cost function with respect to the optimization parameters would equate to zero then.  We generally recommend using quasi-Newton optimizations rather than conjugate gradient ones, because they typically access larger portions of the parameter space.

\textit{How much experimental input is needed?}  We will not give too much advice on the experiments here, interested readers might consult our previous work~\cite{hoerl:17,haberfehlner:17,li:21} to see what worked for us.  Depending on the electron microscope, contamination might play a role and might limit the amount of experimental data.  As has been discussed before, our tomography seems to work best for regions through which swift electrons have traveled.  The reconstruction of blind spots is possible, but the results should be handled with care.

\textit{How to address retardation?}  For surface phonon polaritons the quasistatic approximation works perfectly, but things might be more problematic for the reconstruction of other surface modes, such as surface plasmon polaritons.  In the past we have developed a methodology for surface plasmon tomography including retardation, and have applied the scheme to experimental \textsc{eels} data~\cite{hoerl:15,hoerl:17,haberfehlner:17}.  There we used a bi-orthogonal basis, which shares many features with the resonance modes presented here, but provides no justification for a strictly positive lineshape function.  For this reason, we opted for a compressed sensing optimization that favors expansions with as few modes as possible, where luckily all of them contributed with a positive weight to the loss probability.  In light of our present analysis, we suggest a slight modification of our previous approach.  First, in the retarded case the preferred basis is given by quasinormal modes~\cite{leung:94,lalanne:19,kristensen:20,hohenester.cpc:22}, which have received considerable interest recently.  With these modes we can decompose the dyadic Green's tensor for the full Maxwell's equations in a form similar to Eq.~\eqref{eq:grefl2}, and a tomographic reconstruction should be possible along the lines sketched in the present work.  There remain a number of open issues, such as the proper choice of the reference modes or the consideration of complex mode functions, but we don't foresee any major roadblock.  As a side remark, it is no surprise that the decomposition of the reflected Green's function in terms of resonance modes looks similar in the quasistatic and retarded case: such decompositions are in the spirit of generic singular value decompositions, where the special structure is due to the symmetry property of Green's function originating from the reciprocity theorem of optics.

\textit{How to address membranes and grids?}  Membranes or grids are needed in experiment to support the nanoparticle.  One might wonder about the consequences of such a support in our tomographic reconstruction.  First, modifications of the resonance energies or surface charge distributions of the surface phonon polaritons can be already properly accounted for with the present approach, as has been demonstrated in~\cite{li:21}.  However, in principle also the free-space Green's function of Eq.~\eqref{eq:green2} should be modified to account for the dielectric environment in presence of a support.  This modified Green's function should be used in Eq.~\eqref{eq:eigpot} to propagate away the potentials from the boundary.  It is the resulting modification of the electron-nanoparticle interaction that has to be considered.  Whether this modification has noticeable influence on the results has to be seen.

To summarize, \textsc{eels} tomography has become a successful scheme for reconstructing the three-dimensional photonic environment of nanoparticles with high spatial and energy resolution.  In the past several case studies for plasmonic and photonic nanostructures have provided beautiful results, which would have been hard to achieve with other techniques.  Yet, we feel that there is still enough room for improvements and further investigations.  In this paper we have given an in-depth study of a prototypical nanophotonic system and have demonstrated that tomographic reconstructions work reliably and without major difficulties, at least for systems where the quasistatic approximation can be employed.  We hope that this will motivate more research groups to enter the field, to investigate their systems with the tools presented here, and to continue developing \textsc{eels} tomography with further improvements.

\section*{Acknowledgements}

This project has received funding from the European Union's Horizon 2020 Research and Innovation Program under
grant agreement 823717 (ESTEEM3), from the Austrian Science Fund FWF under project P 31264 and by NAWI Graz.

\begin{appendix}

\section{}\label{sec:ortho}

In this appendix we first derive Eq.~\eqref{eq:eigpotref1}.  We denote the nanoparticle boundary with $\partial\Omega$ and the geometric eigenmodes with $u_k(\bm s)$.  These eigenmodes form a complete set of basis functions and fulfill the orthogonality relation~\cite{ouyang:89,boudarham:12,hohenester:20}
\begin{equation}\label{eq:ortho}
  \oint_{\partial\Omega}\frac{u_k(\bm s)u_{k'}(\bm s')}{4\pi|\bm s-\bm s'|}\,dSdS'=\delta_{kk'}\,.
\end{equation}
Let $u^0_\ell(\bm s_0)$ be the reference basis functions, which are assumed to fulfill a similar orthogonality relation,
\begin{equation}\label{eq:orthoref}
  \oint_{\partial\Omega}\frac{u^0_\ell(\bm s)u^0_{\ell'}(\bm s')}{4\pi|\bm s-\bm s'|}\,dS dS'=
  \delta_{\ell\ell'}\,.
\end{equation}
This can be always achieved for a set of basis functions using a Gram-Schmidt-type orthogonalization.  As $u^0_\ell(\bm s)$ form a complete basis, we can expand the eigenmodes via
\begin{equation}
  u_k(\bm s)=\sum_\ell\mathbb{Q}_{k\ell}u_\ell^0(\bm s)\,.
\end{equation}
Inserting this expression into Eq.~\eqref{eq:ortho} and using the orthogonality relation of Eq.~\eqref{eq:orthoref}, we then immediately observe that $\mathbb{Q}$ is an orthogonal matrix.  In our computational approach we employ Cayley's parameterization for orthogonal matrices
\begin{equation}\label{eq:cayley}
  \mathbb{Q}=\big(\openone+\mathbb{X}\big)\big(\openone-\mathbb{X}\big)^{-1}\,,
\end{equation}
where $\mathbb{X}$ is a skew-symmetric matrix with $x_{ij}=-x_{ji}$.  Eq.~\eqref{eq:cayley} has the advantage that one can perform the derivative $\nicefrac{\partial\mathbb{Q}}{\partial x_{ij}}$ analytically.  To show this, we start from
\begin{equation}\label{eq:cayley2}
  \frac{\partial\mathbb{Q}}{\partial x}=
  \frac{\partial\mathbb{X}}{\partial x}\big(\openone-\mathbb{X}\big)^{-1}+
  \big(\openone+\mathbb{X}\big)\frac\partial{\partial x}\big(\openone-\mathbb{X}\big)^{-1}\,,
\end{equation}
where for notational clarity we have suppressed the subscripts of $x$.  To evaluate the second term on the right hand side, we differentiate $(\openone-\mathbb{X})(\openone-\mathbb{X})^{-1}=\openone$  with respect to $x$.  After some manipulations this leads to
\begin{displaymath}
  \frac\partial{\partial x}\big(\openone-\mathbb{X}\big)^{-1}=
  \big(\openone-\mathbb{X}\big)^{-1}\frac{\partial\mathbb{X}}{\partial x}
  \big(\openone-\mathbb{X}\big)^{-1}\,.
\end{displaymath}
Insertion into Eq.~\eqref{eq:cayley2} gives
\begin{displaymath}
  \frac{\partial\mathbb{Q}}{\partial x}=
  \frac{\partial\mathbb{X}}{\partial x}\big(\openone-\mathbb{X}\big)^{-1}+
  \big(\openone+\mathbb{X}\big)\big(\openone-\mathbb{X}\big)^{-1}\frac{\partial\mathbb{X}}{\partial x}\big(\openone-\mathbb{X}\big)^{-1}\,.
\end{displaymath}
We next use $(\openone+\mathbb{X})(\openone-\mathbb{X})^{-1}=(\openone-\mathbb{X})^{-1}(\openone+\mathbb{X})$, which can be easily proven using $\openone+\mathbb{X}=2\openone-(\openone-\mathbb{X})$ and that both terms on the right hand side commute with $(\openone-\mathbb{X})^{-1}$.  We then arrive at our final expression
\begin{equation}\label{eq:cayleyderiv}
  \frac{\partial Q}{\partial x}=2\big(\openone-\mathbb{X}\big)^{-1}
  \frac{\partial X}{\partial x}\big(\openone-\mathbb{X}\big)^{-1}\,.
\end{equation}
In our optimization of the cost function we express the lineshape function through $L_k=s_k^2$, which guarantees that $L_k$ is always positive.  The optimization algorithm can be significantly accelerated by providing in addition to the value of the cost function also the derivatives with respect to the optimization parameters.  Using Eq.~\eqref{eq:eels3} together with Eq.~\eqref{eq:cayleyderiv}, the derivative of the cost function \eqref{eq:cost} with respect to the parameters $s_k$ and $x_{k\ell}$ of the skew-symmetric matrix can be obtained analytically.  Things are considerably easier for a non-orthogonal matrix where the derivatives with respect to the matrix elements can be performed straightforwardly.

\end{appendix}

redpajama_set_name: RedPajamaArXiv


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redpajama_set_name: RedPajamaCommonCrawl


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Back in 1999, two bohemian Danes decided a funky clothing brand was missing from the fashion scene in Scandinavia and one hazy night, the brand Kronstadt was born. The name itself comes from the Russian town famous for its uprising against the Bolsheviks in 1920's and even led to the birth of a punk band Kronstadt Uprising. Kronstadt stands for change.


June '08 – Match. A person approached by the scammers via Match. This is how she reports Match. Members have to take full responsibility for themselves". See also Internet fraud update , The lottery fraud , The fake job offer fraud , The fake loan offer fraud and The money laundering fraud "This man has been writing to me for some time now and tells me he lives here and is on a job site in Africa, Nigeria, he even sent me flowers and I called the pro-flowers and he used a stolen CC. He had a car crash on the way to his work site. He needed to pay lawyers bills, doctors bills etc.
Businesses often think of the top four social networks — Facebook, Twitter, LinkedIn and MySpace — as the only game in town. But there are hundreds around the globe with millions of users who can be targeted via these networks by businesses. Some of these social networks are general interest, some are niche-lifestyle or interest-specific, and others are widely used internationally and have not yet grown in the United States. Businesses can take advantage of very specific markets and demographics by exploring these 76 social networks and targeting their users to grow their brands in new and exciting ways.
All social networks listed below have a million users or more. Their statistics about membership or monthly traffic were collected from recent public news reports.
Blackplanet Dating Site – If you are looking for a soul mate from the same location, then our site is perfect for you, because you can look up for profiles by your city. .
Diane Blackmon April 3, " They manipulate our history books; the history books are not true; it's a lie. The history books are lying, you need to know that, you must know that. Regarding the American Holiday: Here is what an expedition of European explorers actually found upon their arrival to North American in Five times the size of London or Rome. Great towers and buildings rising from the water.
It ain't a real dating site though so if you really want to go into serious dating you rather use large black dating sites sites like or maybe But this is a great black community forum.
So this consideration should also inform the types of plant foods we prioritise when living off the land. All of these are underground storage organs or USOs. While different, they all serve a similar purpose, the storage of energy so the plant can grow and complete its lifecycle. This energy, preferably stored in the form of starch or other digestible carbohydrate, is what we want to get at. You Must Follow This Simple Rule Because USOs are not visible above ground, we need to take some care to make sure we have collected the correct one, not some underground part of another plant growing nearby: This way, you definitely have the underground parts of the plant you have positively identified, nothing else.
One of these USOs is edible raw. The other two are toxic. All were growing right next to each other, in the same small volume of soil. Cat-tail Greater reedmace, Typha latifolia, and lesser reedmace, Typha angustifolia, are both wetland plants that look very similar. Cat-tails are widely distributed around the Northern Hemisphere. They also occur in South America and parts of Africa, as well as having been introduced to many other territories, where they are often considered an invasive weed.
A stand of Cat-tail growing in a water-filled depression in Ontario, Canada.
These benefits come from milk products such as cheese, butter, cottage cheese, and flavored milk as well. Milk consumption is essential to maintaining good health and is a great source of calcium for all ages. The following are 6 benefits kids and adults gain from drinking milk. Calcium Milk is the best source of calcium for our body.
Blackplanet dating site – Find a man in my area! Free to join to find a man and meet a man online who is single and seek you. Find single man in the US with relations. Looking for romance in all the wrong places? Now, try the right place. Want to meet eligible single man who share your zest for life? Indeed, for those who've tried and failed to find the right man offline, footing can provide.
is tracked by us since April, Over the time it has been ranked as high as 3 in the world, while most of its traffic comes from USA, where it reached as high as 5 position.
You can connect with friends; chat, share photos, videos and music. If you're interested in meeting someone for romance, there's even a separate dating area. BlackPlanet is a social network for black people or anyone else BlackPlanet isn't a dating site but lots of people there are looking for dates and more. It's a good place to meet for romance, friendship, or simply social networking. Though most are African Americans. Joining BlackPlanet You can join for free, chat for free and make dates for free.
Has yahoo messenger and yahoo email which is albertmorgan yahoo. Actually he found me, just 3mths ago. He sent me a lovely message, than I replied we exchanged email address and messenger details. Where we had been chatting and building a relationship, well that is what he led me to believe, we even exchanged phone numbers.
We rang each other many times and he would leave messages on my voice mail how he really missed me and I was everything to him and meant the world to him and soon we would be together. He said he lives in Jacksonville , Florida and was traveling to London for work.
Exploring in Yahoo I at last stumbled upon this site. Studying this information So i am satisfied to express that I have an incredibly just right uncanny feeling I came upon just what I needed. I so much for sure will make sure to don? Terence April 14, at Mischa Barton by Mischa Barton. The pipe smoking is trned as the oldest way of smoking.
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is a community to share and watch short videos on. But when you're done watching and sharing videos on it, you might be considering to close your account.
Dating sites portugal Blackplanet dating site – BlackPlanet. Blackplanet is a social network site for Black and African American people. BlackPlanet is a registered trademark of Community Connect Inc.
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A screenshot of the Style section of BlackPlanet. Its ultimate goal is to bring people together, based generally on common interests. The site maintains channels — subsections of the site — based on broad categories, like music, photography, video and style. BP members congregate at these community areas, increasing the possibility of two perfect strangers meeting one another online, which, of course, is the entire point of social networking. For example, a person on the Dating channel can see the pages of other members.
The slightly less imaginative member can use the "Send Crush" tool, which creates a note through choices made from a drop-down menu "I like your: For coy members, BlackPlanet also features a "Secret Admirer" tool that alerts a potential mate that another member is interested and offers five chances to guess who the admirer may be. The site isn't entirely based on dating and personal encounters, although they make up a substantial portion of the site.
By , The New York Times described it as "unusually successful" despite being "run out of an apartment", having hit five million page views from 50, users. Olympic teammates Tara Lipinski and Michelle Kwan. One writer characterized it "mainly an Internet dating site"  In , Darrell Hamamoto used an interview on the site to attract male talent for his adult film Skin to Skin, using the "unheard of" pairing of an Asian-American male performer with Asian-American woman.
Must Not Include Tags Matched sites that have any tag on this list will not be shown. This feature is useful for filtering out results that have tags you are absolutely not interested in. Types of Results This option lets you specify the types of sites to show. If you want to only see domains www.


This research addresses the challenges of the lack of non-invasive methods and poor spatiotemporal resolution associated with monitoring biogeochemical activity central to bioremediation of subsurface contaminants. Remediation efforts often include growth of biofilm to contain or degrade chemical contaminants, such as nitrates, hydrocarbons, heavy metals, and some chlorinated solvents. Previous research indicates that nuclear magnetic resonance (NMR) is sensitive to the biogeochemical processes of biofilm accumulation. The current research focuses on developing methods to use low-cost NMR technology to support in situ monitoring of biofilm growth and geochemical remediation processes in the subsurface. Biofilm was grown in a lab-scale radial flow bioreactor designed to model the near wellbore subsurface environment. The Vista Clara Javelin NMR logging device, a slim down-the-borehole probe, collected NMR measurements over the course of eight days while biofilm was cultivated in the sand-packed reactor. Measured NMR mean log T2 relaxation times decreased from approximately 710 to 389 ms, indicating that the pore environment and bulk fluid properties were changing due to biofilm growth. Destructive sampling employing drop plate microbial population analysis and scanning electron and stereoscopic microscopy confirmed biofilm formation. Our findings demonstrate that the NMR logging tool can detect small to moderate changes in T2 distribution associated with environmentally relevant quantities of biofilm in quartz sand.
Kirkland CM, Hiebert R, Phillips A, Grunewald E, Walsh DO, Seymour JD, Codd SL, "Biofilm detection in a model well-bore environment using low-field magnetic resonance," Groundwater Monitoring & Remediation 2015 Fall 35(4) 36–44.


The healthcare messaging app, Forward Health, enables clinician-to-clinician communication and information sharing in a secure and compliant way. This healthcare startup was founded by Barney Gilbert, Lydia Yarlott, and Philip Mundy in 2016, and it raised $3.9M led by Harry Stebbings of the podcast The Twenty Minute VC and Stride.VC. Other participants in this round included Albion Capital Group, Fred Destin (founder of Stride.VC), Jay Desai, Melissa Morris, and Christoph Ruedig. Forward Health has now raised a total of $5.2M over two rounds.


as I'm dragged into darkness to be sliced in the shadows.
And if you can't do it tonight, when can you do it?
Same to you, Damien. It exhausts me travelling 100 kilometers to the next departement. Shanghai would kill me!
Happy Halloween & Happy Samhain!


For all you non-military people it's basically checking out of the base. The military members get a list a mile long of things they need to do before they're able to leave.
It's a giant headache, let me tell you.
We had to head off to get a date for our car to be shipped out. Then we had to schedule when the movers would come pick up all our stuff.
Tommy was patient. And because he was patient he was able to pick out a small toy when we stopped in the BXtra. He decided on a small Chewbacca and Clone Trooper.
Going to the BXtra is never fun when I go with Tom. Why? Because if I pick up a toy he's instantly on my side going, "He doesn't need it." Sometimes I don't even plan on buying it, I'm just looking and I have itchy fingers that have to push every button that I see. So if I go down an aisle you might all of a sudden hear a bunch of toys going off at once.
If I see a button, I must push.
They had this singing Pablo from The Backyardigans and of course I had to push it. Pablo said quite cheerfully, "Sing with me!" and Tommy blinked right at him and said in a loud voice, "NO Pablo."
We also did our grocery shopping. And guess what? They had a new Ben and Jerry's flavor.
Mind you, England can take awhile to get the new things from the States. So what is new to us has probably been around in the States for months, possibly even years.
They had a flavor called Peanut Butter Cup.
So of course I had to try that.
Then they had this new ice cream from Breyers. Called Cyclone. The flavor was Chocolate Fudge Brownie. Apparently the Cyclone brand has more chunks than other flavors.
I haven't tried that one yet.
I did try the Peanut Butter Cup. Not as good as I thought it would be but it's edible.
Plus, remember, it's not a pickle and Waffle (fetus) likes pickles.
Only now he's moved on to Peppercinis. They're these spicy things that are delicious. I can eat an entire jar and Tom will watch me in awe and then wave a hand in front of his face, claiming that they stink.
Oh, and I forgot to mention, on Friday Jennifer and I went to see John Tucker Must Die. It was okay. I might rent it when it comes out and watch it with Tom...he might find it appealing..but then never again.
I mean those teenager films, they're all the same.
And there is always a climax at the end where a student talks to a lot of classmates about the importance of being yourself..blah blah blah.
Yes, it's been DONE before you stupid screenwriters.
They did it in Mean Girls. Where Lindsay Lohan's character talked at Homecoming about how everyone was important.
They did it in Never Been Kissed when Drew Barrymore's character talked in front of the Prom.
There is no originality left in the world. A character in Lady in the Water said that and it's so true.
I'm off to call Sky and cancel their services. Let's hope it's easy enough.
"Yes write to this address stating why you want to cancel."
Just cancel me over the phone as they do in the wonderful world called America.


Embroidered Indian braid with red flowers with a orange touch. Very decorative ! Perfect for customizing your creations ! You can, for example, decorate a nice pocket, the edge of sleeves on a jacket or on a curtain.


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POETRY > Epic
The Madman: His Parables and Poems
by Kahlil Gibran
Khalil Gibran (January 6, 1883 - April 10, 1931) was a Lebanese-American writer, poet, visual artist and Lebanese nationalist. Though born a Maronite, Gibran was influenced not only by his own religion but also by Islam, and especially by the mysticism of the Sufis. His knowledge of Lebanon's bloody history, with its destructive factional struggles, strengthened his belief in the fundamental unity of religions, which his parents exemplified by welcoming people of various religions in their home. Connections and parallels have also been made to William Blake's work, as well as the theological ideas of Walt Whitman and in Ralph Waldo Emerson such as reincarnation and the Over-soul. Themes of influence in his work were Islamic/Arabic art, European Classicism (particularly Leonardo Da Vinci) and Romanticism (Blake and Auguste Rodin), the pre-Raphelite Brotherhood, and more modern symbolism and surrealism. Gibran had a number of strong connections to the Bah???'??? Faith starting around 1912. One of Gibran's acquaintances, Juliet Thompson, reported several anecdotes relating to Gibran. She recalled Gibran had met 'Abdu'l-Bah???, the leader of the religion at the time of his journeys to the West Gibran was unable to sleep the night before meeting him in person to draw his portrait in April 1912 on the island of Manhattan. Gibran later told Thompson that in 'Abdu'l-Bah??? he had "seen the Unseen, and been filled". Gibran began work on the book The Prophet, in 1912 when "he got the first motif, for his Island God" whose 'Prometheus exile shall be an Island one."In 1928, after the death of `Abdu'l-Bah???, at a viewing of a movie of `Abdu'l-Bah???, Gibran rose to talk and proclaimed in tears an exalted station of `Abdu'l-Bah??? and left the event weeping still. (wikipedia.org) Read Less
Khalil Gibran (January 6, 1883 - April 10, 1931) was a Lebanese-American writer, poet, visual artist and Lebanese nationalist. Though born a Maronite, Gibran was influenced not only by his own religion but also by Islam, and especially by the mysticism of the Sufis. His knowledge of Lebanon's bloody history, with its destructive factional struggles, strengthened his belief in the fundamental unity of religions, which his parents exemplified by welcoming people of various religions in their home. Connections and parallels ... Read More
Religion > Spirituality
Literary Collections > American
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New UFS Chancellor confers his first degrees
The new Chancellor of the University of the Free State (UFS), Dr Khotso Mokhele, conferred the first degrees since he has taken up the position as Chancellor. The UFS awarded 266 degrees and 20 doctorates to students from the Faculties of the Humanities, Health Sciences, Education, Law and Theology during its Spring graduation ceremony on the Main Campus in Bloemfontein today. Pictured at the ceremony, from the left, are: Dr Mokhele and Prof. Jonathan Jansen (Rector and Vice-Chancellor of the UFS).


Graduate students and postdocs: You now have official permission from the U.S. government to attend that career event.
That's the message of a recent policy clarification from the Council on Financial Assistance Reform, which was created in 2011 by the U.S. Office of Management and Budget (OMB). The clarification states that graduate students and postdocs hold "dual roles" as both trainees and employees. As a trainee, not just an employee, you are authorized to spend time away from the lab to develop other important skills, even when you are supported by federal funds. We've included the OMB language, from this document, in a box at the end of this article, for your reference. Feel free to send a link to your adviser; apparently some advisers still aren't convinced.
Here's the backstory: University administrators (and doubtless some graduate and postdoctoral advisers) were eager to provide career-development opportunities for their graduate students and postdocs. But some advisers—principal investigators (PIs)—worried that if they allowed their advisees to attend, they'd be breaking the law; in their view, those grant-supported trainees were being paid to do research. To address their concerns, an interagency working group involving the National Institutes of Health (NIH), the National Science Foundation, and several other agencies approached OMB: Would they be willing to clarify the policy? OMB readily agreed, and the statement was released on 29 August.
The dual-role clarification is important because training to be a professional scientist—whether you aspire to an academic position or some other kind of job—requires more than just time at the bench. Aspiring academic scientists need to learn to write grants and to teach. Aspiring science journalists need to practice interviewing people for stories. Aspiring tech-transfer specialists, and those who plan to work in industry in business-development roles, must master a large body of knowledge in business and law. You can't get those skills at the bench.
OMB's statement "sends a clear signal that NIH is recognizing that this is a training period and that training is more than just doing work in a laboratory or driving a research project forward," says Naomi Rosenberg, dean of the Sackler School of Graduate Biomedical Sciences at Tufts University, Boston.
Vanderbilt University postdoc Lindsey Morris, who has participated in some of the university's NIH-funded Broadening Experiences in Scientific Training (BEST) career-development programs, knows first-hand the importance of such training, which she did not receive in graduate school. After finishing her Ph.D., she wanted to work at a pharmaceutical company. She got a few interviews, which she thought went well, but no job offers. So, over the next 6 years, she worked as a research assistant in a diabetes lab, a process engineer at a paper mill, and a research associate in a lab studying diabetes and bone disease. Finally, she returned to academia for a postdoc.
Morris sees the clarification as a positive sign because it "shows that NIH knows this is an issue, that postdocs and grad students are not getting specific instruction in soft skills such that they can transition successfully to their next jobs"—whatever those jobs may be.
"I think that it resonates because it comes from the funding source," Hyman says. Others echoed this sentiment, emphasizing the role funding will play as a motivating force for PIs who might at first be hesitant to encourage lab members to pursue career development opportunities. Mathur says that study sections should make funding dependent in part upon whether submitted grants include appropriate training plans and opportunities; that would help convince even resistant PIs to at least consider these issues, she says.
The Uniform Guidance states; "For non-Federal entities that educate and engage students in research, the dual role of students as both trainees and employees contributing to the completion of Federal awards for research must be recognized in the application of these principles." Staff in postdoctoral positions engaged in research, while not generally pursuing an additional degree, are expected to be actively engaged in their training and career development under their research appointments as Post-Docs. This dual role is critical in order to provide Post-Docs with sufficient experience and mentoring for them to successfully pursue independent careers in research and related fields.
Does 200.400(f) require recognition of the dual role of postdoctoral staff appointed on research grants as, both trainees and employees, when appointed as a researcher on research grants?
Yes, the Uniform Guidance 200.400(f) requires the recognition of the dual role of all pre and post-doctoral staff, who are appointed to research positions with the intent that the research experience will further their training and support the development of skills critical to pursue careers as independent investigators or other related careers.
Neither Pre-Docs or Post-Docs need to be specifically appointed in 'training' positions to require recognition of this dual role. The requirements and expectations of their appointment will support recognition of this dual role per 200.400(f).
Rachel Bernstein is a staff writer for Science Careers.


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Kilkis (gr. Δήμος Κιλκίς, Dimos Kilkis) – gmina w Grecji, w administracji zdecentralizowanej Macedonia-Tracja, w regionie Macedonia Środkowa, w jednostce regionalnej Kilkis. W 2011 roku liczyła 51 926 mieszkańców. Powstała 1 stycznia 2011 roku w wyniku połączenia dotychczasowych gmin: Kilkis, Krusi, Cherso, Pikrolimni, Muries, Galikos i Doirani. Siedzibą gminy jest Kilkis.

Przypisy

Gminy w Grecji
Region Macedonia Środkowa

redpajama_set_name: RedPajamaWikipedia


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Hello everyone, welcome to the Music page!
Hello and welcome to our Music page!
On here you'll find out all about what we're getting up to at St Stephen's Church of England Primary to showcase our musical talents! At St Stephen's, we use a Music scheme throughout school called Charanga. This is used as part of fantastic teacher knowledge and skills to provide an all inclusive and enriched Music programme, progressing through all levels of attainment.
The Benefits of Music
Music is not only something that can be loved and appreciated by everyone through enjoyment, but it also has lots of educational benefits, particularly for children. Please find a useful website link below and some extracts from this, which highlight just how beneficial Music can be in supporting a young person's academic progress.
-Children who study Music tend to have larger vocabularies
-Children in schools with an enriched Music programme scored higher in Mathematics and English
-Studying Music could have cognitive benefits similar to those evident in bilingual children and can support this
-Everyday listening skills are better in those children who have a more enriched Music programme
-Children with learning needs or dyslexia who may lose focus more easily, tend to benefit greatly from Music lessons
-Children with better Music pedagogy and experience tend to have better memory
https://www.nammfoundation.org/articles/2014-06-09/how-children-benefit-music-education-schools?gclid=EAIaIQobChMI7pL8wqOl7AIVlO3tCh22EgFLEAAYASAAEgLCO_D_BwE
Click here to look at an overview of the scheme
Click below to see our overview of Music this academic year at St Stephens:
Music Overview
Music Overview EYFS-Y6.docx
Our Music Policy
Music Policy 2021.doc
2020 Music in School!
As we all know, coming back to school has presented many changes for both the staff and children, and also for Music. At the moment, our school is still alive with the sound of Music, and we are still listening to our most loved and favourite songs throguhout the day. Mrs Caswell from Bolton Music Service is still running Music CPD lessons on Wednesday afternoons, which the children are thoroughly enjoying. Year 4 still continue with their Musical instrument lessons where they have the opportunity to learn the Ukulele.
We have also been using Music as a way to relax and find time to unwind.
Below are some useful Music websites that children can access at home to develop their Musical skills, or to help relax.
Music Link-
https://pbskids.org/games/music/
In 2019 we were delighted to be able to welcome back Pop UK! Children in all year groups were able to learn new songs, enjoying singing them and even recording them to make an album! The school was alive with music and the children really enjoyed this experience. Children were able to take part in afternoon and evening concerts, where they sang and performed their new music. Thank you to all the families and friends that came along to join in with our concerts and enjoying the children's music. We are still enjoying our Pop UK songs that the children recorded!
We can't wait to welcome Pop UK back in the near future!
Worship and Singing Practise
We use some fantastic songs as part of our worship in school. Along with producers of various collections that we use, we subscribe to an online source called Fischy.com. The songs have religious and moral content which the children love to sing. The children learn signs along with the songs to help them remember and enjoy the words. The Fischy Music leader Stephen Fischbacher came to school to lead a two day workshop teaching the children new songs which resulted in a performance for the parents. In addition to singing, we listen to a wide range of music on entry and exit to our Worship time with God. This may be quiet and reflective or lively and fun-loving. Each week, we practise our songs in singing practise.
You can visit the Fischy Music website and sample some of the songs using the link below:
https://www.fischy.com/
Fantastic support from the Bolton/Blackburn with Darwen Music Service
We continue our fantastic partnership with the music service, who have been supporting our teachers professional development in Music sessions. Angela Caswell is currently working with our class teachers in delivering our online 'Charanga' music scheme, to ensure the highest quality of Music is being delivered to our children.
Throughout school, our children are learning to sing, play instruments, compose, and appreciate all genres of music from classical to rap. Take a look at the scheme through the link below:
https://charanga.com/site/
Pop UK Concert!
Useful websites-
BBC songs and nursery rhymes Songs and nursery rhymes to watch and sing along to
Fischy Music Sample some of our school worship songs
BBC Bitesize Music Year KS1
https://www.bbc.co.uk/bitesize/subjects/z7tnvcw
BBC Bitesize Music KS2-
https://www.bbc.co.uk/bitesize/subjects/zwxhfg8