text:
\section{Introduction \label{sec1}}
Polar materials have become of high interest in the field of nanophotonics due to their ability to support phonon polaritons, quasi-particles which result from the coupling between electromagnetic waves and crystal lattice vibrations \cite{Zoo2016,Mills1974} with a characteristic wavelength lying in the mid-infrared region. These quasi-particles can enhance the electromagnetic field deep below the diffraction limit with large quality factors compared to infrared plasmons \cite{Hillenbrand2002,DeAbajo2016,Caldwell2019}, making them promising building blocks for infrared nanophotonics applications \cite{Zubin2014,Caldwell2015,Koppens2017,Caldwell2019-2}.

One interesting 2D polar material is hexagonal boron nitride (h-BN) because of its high quality phonon polaritons and the easy preparation of the single atomic layers made by exfoliation \cite{Basov2014,Caldwell2014,Raschke2015,Hillenbrand2017,Walker2017,Ambrosio2018}. Besides being widely used in heterostructures \cite{Caldwell2015-2}, h-BN is emerging by itself as a versatile material offering novel optical and electro-optical functionalities. The crystal layer structure that constitues h-BN, mediated via van der Waals forces, produces an uniaxial optical response of the material. This implies that the dielectric function of h-BN needs to be described by a diagonal tensor $\tensor{\varepsilon}$ with two principal axes \cite{Caldwell2014,Raschke2015}: $\varepsilon_x=\varepsilon_y=\varepsilon_{\bot}$ and $\varepsilon_z=\varepsilon_{\parallel}$. When $\text{Re}(\varepsilon_{\parallel}) \cdot \text{Re}(\varepsilon_{\bot})<0$, phonon polaritons can propagate inside the material and exhibit a hyperbolic dispersion \cite{Kivshar2013,Zubin2014}, that is, the relationship between the different components of the polariton wavevector $\vect{k}(\omega)=(k_x,k_y,k_z)$ traces a surface in momentum space which corresponds to hyperboloids. For h-BN, one can find two energy bands (the Reststrahlen bands) where one of the principal components of the dielectric tensor is negative. Each Reststrahlen band is defined by the energy region between the transverse and longitudinal optical phonon energy, TO and LO, respectively ($\text{TO}_{\bot}$ and $\text{LO}_{\bot}$ for the upper Reststrahlen band and $\text{TO}_{\parallel}$ and $\text{LO}_{\parallel}$ for the lower Reststrahlen band, see Fig. 1). On the other hand, when $\text{Re}(\varepsilon_{\parallel}) \cdot \text{Re}(\varepsilon_{\bot})>0$, the isofrequency surfaces traced by the polariton wavevector in momentum space are ellipsoids.
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.92]{fig1.png}
\caption{Real parts of the components of the h-BN dielectric function. The shaded red area marks the lower Reststrahlen band and the grey area the upper Reststrahlen band. Insets illustrate the elliptic and the hyperbolic (Type I and Type II) isofrequency surfaces and the crystal lattice structure of h-BN.}
\label{fig1}
\end{center}
\end{figure}
\vspace{-5.2mm}

Figure \ref{fig1} depicts $\varepsilon_{\bot}$ and $\varepsilon_{\parallel}$ (see appendix \ref{appendA} for expressions and parameters of the dielectric tensor components), which represent the in-plane and out-of-plane dielectric components of h-BN, respectively. The energy range in Fig. \ref{fig1}, shaded in red, corresponds to the lower Reststrahlen band (94.2-102.3\,meV) where the real part of the out-of-plane permittivity is negative, leading to isofrequency surfaces in the form of two-sheet hyperboloids (inset Type I). The energy region shaded in grey corresponds to the upper Reststrahlen band (168.6-200.1\,meV), where the real part of the in-plane permittivity is negative and the isofrequency surfaces correspond to one-sheet hyperboloids (inset Type II).

Hyperbolic phonon polaritons excitable on h-BN within the range of 90 – 200 meV might be a key to many novel photonic technologies relying on the nanoscale confinement of light and its manipulation.  As a result, efficient design and utilization of h-BN structures require spectroscopic studies with adequate spatial resolution. This can be provided, for instance, by electron energy loss spectroscopy (EELS) using electrons as localized electromagnetic probes. Recently, instrumental improvements in EELS performed in a scanning transmission microscope (STEM-EELS) allowed to spatially map phonon polaritons \cite{,Crozier2014} and hyperbolic phonon polaritons in h-BN\cite{,Konecna2017,Ramasse2018}. The focused electron beam of an electron microscope has thus become a suitable probe to access the spectral information of low-energy excitations in technologically relevant materials, with nanoscale spatial resolution. Thus, EEL spectra in phononic materials can be of paramount importance to reveal the properties of phonon polariton excitations.

In this work we first show that a fast electron traveling through bulk h-BN can excite volume phonon polaritons inside and outside the h-BN Reststrahlen bands. Our analysis reveals that the excitation of the volume polariton modes is strongly dependent on the electron velocity and also on the orientation between the electron beam trajectory and the h-BN optical axis. We then study the formation of wake patterns in the field distribution induced by the electron beam at h-BN. Our methodology allows us to connect the excitation of these wake fields with the different electron energy loss mechanisms experienced by the fast electron in the medium: (i) excitation of phonon polaritons or (ii) Cherenkov radiation. We also discuss the emergence of asymmetric wake patterns exhibited by the induced electromagnetic field when the electron beam trajectory sustains an angle relative to the h-BN optical axis. Finally, in the last two section of the paper we show that a fast electron beam interacting with a semi-infinite h-BN interface excite Dyakonov surface phonon polartions within the h-BN upper Reststrahlen band. We further demonstrate that the probing electron traveling above the h-BN in aloof trajectories excites volume phonon polaritons (remotely activation). All these findings offer a way to steer and control the propagation of the polaritonic waves and reveal the importance of the anisotropic optical response of the material in the EELS analysis.
\section{Excitation of infrared bulk modes in h-BN by a focused fast electron beam\label{sec2}}
\subsection{Bulk modes in h-BN}
According to Maxwell's equations in momentum-frequency ($k-\omega$) space, the dispersion relation for a wave propagating in the volume of an anisotropic material can be found from the following relationship: \cite{Eroglu2010}
\begin{equation}
\label{2.a}
\text{det}[\tensor{\bf G}^{-1}(\vect{k};\omega)]=\text{det} \left[\vect{k} \otimes \vect{k} - k^2 \tensor{\bf I} + k^2_0 \tensor{\varepsilon}\right]=0,
\end{equation}
where $\tensor{\bf G}^{-1}$ is the inverse of the Green's tensor, $\vect{k}(\omega)=(k_x,k_y,k_z)$ is the wavevector of the wave, $k_0=\omega/c$ is the magnitude of the wavevector in vacuum, $c$ is the speed of light, $\text{det}[\tensor{\bf x}]$ stands for the determinant of a matrix, $\otimes$ is the tensor product and $\tensor{\bf I}$ is the identity tensor. Particularly, for an uniaxial medium, the dielectric response can be described in tensor form as $\tensor{\varepsilon}(\omega)= \text{diag}[\varepsilon_{\bot},\varepsilon_{\bot},\varepsilon_{\parallel}]$. For this case, two solutions (modes) arise from Eq. \myref{2.a}, yielding the dispersion relation for ordinary waves
\begin{equation}
\label{2.b}
k^2= k^2_0 \varepsilon_{\bot},
\end{equation}
and the dispersion relation for extraordinary waves
\begin{equation}
\label{2.c}
\frac{k^2_x+k^2_y}{\varepsilon_{\parallel}}+\frac{k^2_z}{\varepsilon_{\bot}}=k^2_0.
\end{equation}
Equation \myref{2.b} represents concentric spheres in \textit{k}-space for a given energy $\hbar\omega$ (with $\varepsilon_{\bot}>0$), while Eq. \myref{2.c} represents hyperboloids or ellipsoids in the reciprocal space depending on the sign of the dielectric components $\varepsilon_{\parallel}$ and $\varepsilon_{\bot}$. Thus, we conclude that the isofrequency surfaces of the polariton wavevector $\vect{k}(\omega)$ in momentum space (for an uniaxial medium) represent geometrically spheres, ellipsoids or hyperboloids. For h-BN, the insets in Fig. \ref{fig1} depict the isofrequency surfaces for each energy region inside and outside the Reststrahlen bands.
\subsection{Electron energy loss probability}
Fast electron beams can couple to bulk polaritonic modes sustained in anisotropic media. We can observe this by analyzing the energy losses experienced by the electron when traveling in such media. Electron energy losses, $\Delta E_{\text{EELS}}$, can be calculated within classical electrodynamics as the work performed by the induced electromagnetic field, $\vect{E}^{\text{ind }}(\vect{r};t)$, on the probing electron\cite{Ritchie1957,Ritchie1985,DeAbajo2010-2,DeAbajo2019,Hohenester}
\begin{equation}
\label{2.d}
\Delta E_{\text{EELS}} = e \int  \text{d}\vect{r}_{e} \cdot \vect{E}^{\text{ind }}(\vect{r}_{e};t),
\end{equation}
where the integration is performed along the electron beam trajectory ${\vect{r}}_{e}(t)$, $e$ is the elementary charge, and $\vect{E}^{\text{ind}}(\vect{r};t)$ is evaluated along ${\vect{r}}_{e}(t)$. Notice that we approximate the electron beam as a classical point charge. The high-energy currents used in typical EELS experiments justify this approximation \cite{Richtie1981,Richtie1988,Rivacoba2014}. If we Fourier transform $\vect{E}^{\text{ind}}(\vect{r};t) \mapsto \vect{E}^{\text{ind}}(\vect{r};\omega)$ in Eq. \myref{2.d}, the electron energy losses can be written as
\begin{align}
\label{2.d.1}
\Delta E_{\text{EELS}} &= \frac{e}{2\pi} \int  \text{d}\vect{r}_{e} \cdot \int_{-\infty}^{\infty} \text{d}\omega\, \vect{E}^{\text{ind}}(\vect{r}_{e};\omega)\, e^{-i\omega t}\\
\nonumber
&=\int_0^{\infty} \text{d}\omega\, \int \text{d}L\, \hbar\omega \, \Gamma(\omega),
\end{align}
where one identifies the electron energy loss (EEL) probability per unit path, $\Gamma(\omega)$, as
\begin{equation}
\label{2.f}
\Gamma(\omega)= \frac{e}{\pi \hbar \omega} \text{Re} \left[ \vect{E}^{\text{ind}}({\vect{r}}_{e};\omega) \cdot \uvect{v} \, e^{-i\omega t_{e}} \right],
\end{equation}
with $\uvect{v}$ the unit vector in the same direction as the electron velocity $\vect{v}$ and $t_e$ is the time for the electron to travel a distance $\text{d}L$. Hence, to calculate $\Gamma(\omega)$ one needs to know the induced electric field $\vect{E}^{\text{ind}}(\vect{r};\omega)$. We derive below the expressions of the total electric field $\vect{E}^{\text{tot}}(\vect{r};\omega)$ and $\Gamma(\omega)$ for an electron beam trajectory parallel to the optical axis of h-BN, as depicted in Fig. \ref{fig2}.
\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.70]{fig2.png}
\caption{Schematics of the electron traveling through the h-BN with velocity $\vect{v}=v \uvect{z}$ parallel to the h-BN optical axis (\textit{z}-direction).}
\label{fig2}
\end{center}
\end{figure}

It follows from Maxwell's equations that the field produced by the fast electron plus the induced electric field, namely the total electromagnetic field ($\vect{E}^{\text{tot}}(\vect{r};\omega)$) is given by
\begin{equation}
\label{2.g}
\vect{E}^{\text{tot}}(\vect{r};\omega)=-i \frac{\omega}{(2\pi)^3 c^2 \varepsilon_0} \int \text{d}^3\vect{k} \,\, \rho(\vect{k};\omega)\, \tensor{\bf G}(\vect{k};\omega) \cdot \vect{v}\, e^{i\vect{k} \cdot \vect{r}},
\end{equation}
with $\varepsilon_{0}$ the vacuum permittivity and $\rho(\vect{k};\omega)= -2 e \pi \delta(\omega - \vect{k}\cdot\vect{v})$ the charge density of the probing electron. The integration in Eq. \myref{2.g} extends over the whole reciprocal space and the delta function introduced by the charge density assures conservation of energy and momentum. Indeed, one finds that in the non-relativistic limit the energy that the electron with initial velocity $\vect{v}$ transfers to the medium upon loosing momentum $\hbar\vect{k}$ is
\begin{equation}
\label{2.g.1}
\hbar \omega =\frac{(\vect{p}+\hbar \vect{k})^2}{2m}-\frac{p^2}{2m} = \hbar \vect{v}\cdot\vect{k}+\frac{\hbar^2}{2m}k^2,
\end{equation}
with $\vect{p}=m\vect{v}$ the initial momentum of the fast electron. By neglecting  recoil of the incident electron, from Eq. \myref{2.g.1} one arrives to the so-called non-recoil approximation where $\omega=\vect{k}\cdot\vect{v}$. Note that the \textit{z}-component of the wavevector is fixed by $k_z=\omega/v$ when the electron travels in the \textit{z}-direction.

To calculate the bulk loss probability $\Gamma^{\text{bulk}}(\omega)$  experienced by the  fast electron in the anisotropic medium we substitute Eq. \myref{2.g} into Eq. \myref{2.f}. Notice that a fast electron traveling in vacuum looses no energy, this allows to use $\vect{E}^{\text{tot}}(\vect{r};\omega)$ instead of $\vect{E}^{\text{ind}}(\vect{r};\omega)$ in Eq. \myref{2.f}. Using the cylindrical symmetry of the field produced by the fast electron one finds that
\begin{equation}
\label{2.h}
\Gamma^{\text{bulk}}(\omega) = \int_{0}^{k^{\text{c}}_{\bot}} \text{d}k_{\bot} \,  P^{\text{bulk}}(k_{\bot};\omega),
\end{equation}
where
\begin{equation}
\label{2.i}
P^{\text{bulk}}(k_{\bot};\omega) = -\frac{2 e^2 k_{\bot} v}{(2\pi)^3 \hbar c^2 \varepsilon_0 v_z} \int_0^{2\pi} \text{d}\phi \, \text{Im} \left[\uvect{v} \cdot \tensor{\bf G}^* \cdot \uvect{v} \right],
\end{equation}
is the probability for the electron to transfer a transverse momentum $\hbar k_{\bot}$ (to the electron trajectory) upon lossing energy $\hbar\omega$. We will refer to this quantity as the momentum-resolved loss probability. In Eq. \myref{2.i} ${\tensor{\bf G}^*}={\tensor{\bf G}}(k_{\bot},\phi,k_z=\omega/v_z -  \vect{k}_{\bot}\cdot \vect{v}/v_z)$, and $\phi$ is the angle between $\vect{k}_{\bot}$ and the $k_x$-axis, with $\hbar k^{\text{c}}_{\bot}$ the maximum perpendicular momentum of the electrons selected by the collection aperture of the EELS spectrometer.

Particularly, when the electron beam trajectory points out in the same direction as the h-BN optical axis ($\vect{v}=v \uvect{z}$), expressions for $\vect{E}^{\text{tot}}(\vect{r};\omega)$ and thus for $\Gamma^{\text{bulk}}(\omega)$ can be found in a closed form (see appendix \ref{appendB} for the analytical formula of the Green's tensor in uniaxial anisotropic media):
\begin{equation}
\label{2.j}
\vect{E}^{\text{tot}}(\vect{r};\omega)= \frac{e}{2\pi \varepsilon_0} \frac{\omega}{v^2 \gamma_{\bot} \varepsilon_{\bot}} \vect{g}(\vect{r};\omega),
\end{equation}
where
\begin{align}
\label{2.k}
\nonumber
\vect{g}(\vect{r};\omega)&= e^{i\omega z/v} \left[  \frac{i}{\gamma_{\bot} }  K_0\left( \sqrt{\frac{\varepsilon_{\parallel}}{\varepsilon_{\bot}}} \frac{\lvert \omega \rvert}{\gamma_{\bot} v} R \right)  \uvect{z}  \right. \\
& \hspace{4mm} \left. -\, \text{sgn}(\omega) \sqrt{\varepsilon_{\parallel} \varepsilon_{\bot}}  K_1\left( \sqrt{\frac{\varepsilon_{\parallel}}{\varepsilon_{\bot}}} \frac{\lvert \omega \rvert}{\gamma_{\bot} v} R \right) \uvect{R} \right],
\end{align}
is written in cylindrical coordinates $\vect{r}=(\vect{R},z)=(x,y,z)$, $R=\sqrt{x^2+y^2}$, $K_0(x)$, $K_1(x)$ are the zero and first order modified Bessel functions of the second kind, sgn stands for the sign function and  $\gamma_{\bot}= 1/\sqrt{1-v^2 \varepsilon_{\bot}/c^2}$ is the Lorentz factor.

The momentum-resolved loss probability becomes
\begin{align}
\label{2.l}
\nonumber
P^{\text{bulk}}(k_{\bot};\omega) &= -\frac{2 e^2}{(2\pi)^2 \omega^2 \hbar \varepsilon_0} \text{Im} \left\{ \left[ k_0^2 \varepsilon_{\bot} - \frac{\omega^2}{v^2} \right]\right. \\
& \hspace{4mm} \left. \times \frac{k_{\bot}}{\varepsilon_{\parallel}[\varepsilon_{\bot} k_0^2-\omega^2/v^2] - \varepsilon_{\bot} k_{\bot}^2}  \right\},
\end{align}
and, substituting Eq. \myref{2.l} into Eq. \myref{2.h}, one finds that

\begin{align}
\label{2.m}
\nonumber
\Gamma^{\text{bulk}} (\omega) &=  \frac{e^2}{(2\pi)^2 \omega^2 \hbar \varepsilon_0} \text{Im} \left\{ \left[ k_0^2 - \frac{\omega^2}{\varepsilon_{\bot} v^2} \right] \right. \\
& \hspace{4mm} \left. \times\ln \left[\frac{\varepsilon_{\parallel} \varepsilon_{\bot} k_0^2- \varepsilon_{\parallel}\omega^2/v^2 - \varepsilon_{\bot} (k_{\bot}^c)^2}{\varepsilon_{\parallel} \varepsilon_{\bot} k_0^2 - \varepsilon_{\parallel} \omega^2/v^2} \right]  \right\}.
\end{align}
The non-retarded versions of $P^{\text{bulk}}(k_{\bot};\omega)$ and $\Gamma^{\text{bulk}}(\omega)$ can be obtained by setting $k_0$ equal to zero in Eqs. \myref{2.l} and \myref{2.m}.

The spectrum of the momentum-resolved loss probability and the EEL probability provide valuable information which reveals the properties of the modes of the anisotropic material. We thus explore in the following the connection between the dispersion relation of the h-BN excitations in the upper Reststrahlen band with these two quantities.
\subsection{Upper Reststrahlen band}
\begin{figure*}[!ht]
\begin{center}
\includegraphics[scale=0.8]{fig3.png}
\caption{(a) Isofrequency curves for energies inside (195\,meV, red solid line) and outside(160 and 205\,meV, green and black dashed lines) the upper Reststrahlen band plotted for the wavevector $k_z$ versus $k_{\bot}=\sqrt{k^2_x+k^2_y}$. The horizontal blue line represents the momentum $\hbar k_z=\hbar \omega/(0.1c)$ transferred by the fast electron to the polaritons when it travels along the \textit{z}-direction with $v=0.1c$. The blue line is evaluated at energy 195\,meV (caption at the top right of the figure). The black arrows represent the polariton wavevector $\vect{k}(\omega)$, $\theta_{\vect{k}}$ is the angle between $\vect{k}(\omega)$ and the $k_z$-axis, the magenta arrows represent the group velocity $\vect{v}_g$ and the orange arrows the Poynting vector $\vect{S}$. Panel (d) shows a zoom into (a). In (d) the horizontal blue line represents $k_z=\omega/(0.5c)$ . The contour plot (left panel) in (b) shows the momentum-resolved loss probability $P^{\text{bulk}}(k_{\bot};\omega)$ normalized to the maximum value (3\,a.u.) for $v=0.1c$. The right panel in (b) shows the energy loss probability $\Gamma^{\text{bulk}}(\omega)$ obtained by integrating $P^{\text{bulk}}(k_{\bot};\omega)$ over $k_{\bot}$ up to $k^c_{\bot}=0.05\,\mathring{\text{A}}^{-1}$. (e) same as in (b) but considering $v=0.5c$. (c) and (f) depict the real part of the \textit{z}-component of the total electric field induced by the fast electron along the cylindrical coordinates  ($R,z$) for the energy 195\,meV. The field plots are normalized to the maximum value in each case: (c) $1\times10^{-5}\,\text{a.u.}$ and (f) $7.5\times10^{-7}\,\text{a.u.}$ The insets  in (c) and (f) illustrate the electron beam trajectory and orientation of the h-BN crystal planes.}
\label{fig3}
\end{center}
\end{figure*}

In the following we address the electron energy losses in h-BN  and the connection of these losses with the isofrequency surfaces of the material. We first show in Fig. \ref{fig3}a the isofrequency curve of a h-BN phonon polariton for an energy in the upper Reststrahlen band (red curve). We chose 195\,meV as a representative value of this band. When a fast electron beam is used to probe these excitations in the medium, the velocity of the electron determines the momentum transfer, as $\vect{k}\cdot\vect{v}=\omega$ (Eq. 8). If the electron is traveling along the \textit{z}-direction, then $k_z=\omega/v$ (blue horizontal line in Fig. \ref{fig3}a). Following Eq. \myref{2.c}, this also sets the value of the $\hbar k_{\bot}$ momentum component ($k^2_{\bot}=\varepsilon_{\parallel}k^2_0-\varepsilon_{\parallel} k^2_z/\varepsilon_{\bot}$) of the excited phonon polariton.

The intersections between $k_z=\omega/v$ and the isofrequency curves in the upper Reststrahlen band establish a relationship between the energy $\hbar\omega$ of the hyperbolic phonon polariton and its perpendicular momentum component $\hbar k_{\bot}$. In the left panel of Fig. \ref{fig3}b we plot this relationship (blue dashed line) and the momentum-resolved loss probability $P^{\text{bulk}}(k_{\bot};\omega)$ (light blue-yellow contour plot) for $v=0.1c$. We note that the highest values of $P^{\text{bulk}}(k_{\bot};\omega)$ coincide with the blue dashed line and its asymptotic behavior approaches $\text{LO}_{\bot}$ for large $k_{\bot}$. This demonstrates that electron energy losses in the upper band are due to phonon polariton excitations. We confirm this by integrating $P^{\text{bulk}}(k_{\bot};\omega)$ over $k_{\bot}$ up to a cutoff momentum $\hbar k^c_{\bot}$, which yields the EEL probability $\Gamma^{\text{bulk}}(\omega)$ (right panel of Fig. \ref{fig3}b). A clear peak can be observed at the longitudinal optical phonon. This energy loss peak is slightly asymmetric with a broader tail inside the Reststrahlen band compared to that outside the band.  Importantly, at energies above $\text{LO}_{\bot}$ no losses are found. This can be understood with the help of the isofrecuency curves in Fig. \ref{fig3}a. For instance, at energy 205\,meV (black dashed line, above the upper Reststrahlen band) the ellipse does not intersect the blue horizontal line and therefore there is no excitation above the upper band. For energies below $\text{TO}_{\bot}$, the ellipses may intersect or not the blue horizontal line of $k_z$ depending on the particular energy. For instance, at an energy of 160\,meV (green dashed line, below the upper Reststrahlen band in Fig. \ref{fig3}a) the ellipse does not cut $k_z=\omega/(0.1c)$ and therefore there is no excitation induced in that case. However, for lower energies the isofrequency surfaces can cut the $k_z$ line, and therefore an anisotropic dielectric mode can be excited (tail below 170\,meV in Fig. \ref{fig3}b). We learn from this analysis that the excitation of the phonon polariton modes close to the upper Reststrahlen band is highly dependent on the topology (hyperbolic or elliptic) of the isofrequency surfaces.

The dependency of phonon polaritons excitation on the isofrequency surface allows to control the polaritonic modes as we discuss now in Fig. \ref{fig3}c, where we show the real part of the \textit{z}-component of the total electric field at $\hbar\omega= 195\,\text{meV}$ (representing the energy within the hyperbolic dispersion regime in the upper Reststrahlen band), induced by a fast electron with velocity $v=0.1c$. A schematic representation of such electron beam trajectory is displayed in the inset of Fig. \ref{fig3}c.  We observe two important features: the formation of a wake pattern and an oscillatory behavior of the field in the \textit{z}-direction. This spatial periodicity is connected with the parallel momentum component ($\hbar k_z=\hbar \omega/v$) transferred by the electron since the observed wavelength along the \textit{z}-axis is $\lambda_z=2\pi/k_z$. This implies that the wavelength $\lambda_z$ decreases with increasing energy of the phonon polariton. Furthermore, the direction of the wake pattern is governed by the polariton phase velocity ($\vect{v}_p$ parallel to $\vect{k}(\omega)$, black arrow). The outward direction (relative to the electron beam trajectory) of the wavefronts is determined by the sign of the radial component of $\vect{v}_p$ relative to the radial component of the energy flow (given by the Pointing vector $\vect{S}=\vect{E}\times\vect{H}$ parallel to the group velocity $\vect{v}_g=\nabla_{\vect{k}} \omega$\cite{Felsen1994,Galyamin2011,Hillenbrand2015,Landau,Amnon}, magenta arrow). We recognize in Fig. \ref{fig3}a that the group and the phase velocities are nearly perpendicular, and their projection onto the radial axis are parallel, leading to a wave propagating away from the electron beam trajectory (positive phase and positive group velocity with respect to the energy propagation direction). It is worth noting that the projection of the group and the phase velocities onto the beam trajectory direction (\textit{z}-direction) leads to positive positive phase and negative group velocities relative to $S_z$ (Fig. \ref{fig3}a).

As pointed out, for each energy $\hbar\omega$, the velocity of the fast electron determines (primarily) the polariton wavevector parallel to the beam trajectory, $k_z$, and consequently the perpendicular wavevector $k_{\bot}$ (according to Eq. \myref{2.c}). To emphasize the velocity dependency, we perform the same analysis  (Fig. \ref{fig3}d-f) as in Fig. \ref{fig3}a-c but increasing the electron velocity to $v=0.5c$. In Fig. \ref{fig3}d a zoom into the isofrequency curve of Fig. \ref{fig3}a is presented, together with the value $k_z$ (horizontal blue line) determined by the electron velocity $v= 0.5c$. The increase of the electron velocity leads to the excitation of 195\,meV polaritons with reduced momentum (determined by the intersection of the blue horizontal line and the red isofrequency curve). By calculating the momentum-resolved loss probability $P^{\text{bulk}}(k_{\bot};\omega)$ (left panel of Fig. \ref{fig3}e) and the EEL probability $\Gamma^{\text{bulk}}(\omega)$ (right panel of Fig \ref{fig3}e) we find the same behavior as in Fig. \ref{fig3}b for $v=0.1c$, except for a one order of magnitude reduction in both $k_{\bot}$ and the value of the loss probability.

The differences in the properties of the phonon polaritons launched by the fast electron at both electron velocities are distinguishable in Fig. \ref{fig3}f, where we show the real part of  the \textit{z}-component of the total electric field induced by the fast electron with $v=0.5c$ at energy 195\,meV. The spatial period $\lambda_{z}$ of the polariton is longer compared to that in Fig. \ref{fig3}c as a result of the increase in the electron velocity (smaller $\hbar k_z$ transferred). Interestingly, the direction of the wake field is quite similar to that of panel \ref{fig3}c. This behavior is a specific feature of hyperbolic polaritons since the intersection of the blue line both for $v=0.1c$ and $v=0.5c$ occur at the asymptote of the hyperbola which results in polariton wavevectors that have very similar propagation direction but different absolute values.

\subsection{Lower Reststrahlen band}
In Fig. \ref{fig4}a we show the isofrequency curve of h-BN phonon polaritons for an energy in the lower Reststrahlen band (red line). Note that the hyperbolas are rotated by $90^{\circ}$ as compared to the upper Reststrahlen band (see Fig. \ref{fig3}a and \ref{fig3}d). However, the momentum $\hbar k_z$ transferred by the fast electron to the phonon polaritons is still given by the crossing of the hyperbolas with the horizontal blue line (representing $k_z =\omega/v$ for $v=0.1c$ in Fig. \ref{fig4}a). From Eq. \myref{2.c} we obtain the polariton perpendicular momentum $\hbar k_{\bot}$, which is shown in Fig. \ref{fig4}b as a function of energy $\hbar\omega$ (dashed blue curve). We also plot the momentum-resolved loss probability $P^{\text{bulk}}(k_{\bot};\omega)$ for energies within the lower Reststrahlen band. Notice that the highest values of $P^{\text{bulk}}(k_{\bot};\omega)$ (red and yellow colors in the contour plot) coincide perfectly with the blue dashed curve, demonstrating that the electron energy losses in the lower band are also governed by polariton excitations. However, in contrast to the upper band, we find that the dashed blue curve has a negative slope, $\text{d}\omega/\text{d}k_{\bot} < 0$, indicating that the group and the phase velocities are antiparallel (have opposite sign) along the radial direction. We will show below with Fig. \ref{fig4}c that the phase velocity in the radial direction is indeed antiparallel (negative) relative to the Poynting vector (energy flow) while the group velocity in the radial direction is parallel (positive), which is a consequence of the phase and group velocity vectors being perpendicular to each other and rotated by $90^{\circ}$ degrees compared to the upper Reststrahlen band.
\begin{figure*}[!ht]
\begin{center}
\includegraphics[scale=0.8]{fig4.png}
\caption{(a) Isofrequency curves for energies inside (100\,meV, red solid line) and outside (92.5 and 105\,meV, green and black dashed lines) the lower Reststrahlen band plotted for the wavevector $k_z$ against $k_{\bot}=\sqrt{k^2_x+k^2_y}$. The horizontal blue line represents the momentum $\hbar k_z=\hbar \omega/(0.1c)$ transferred by the fast electron to the polaritons when it travels along the \textit{z}-direction with velocity $0.1c$. The blue line is evaluated at energy 100\,meV. The black arrows represent the polariton wavevector $\vect{k}(\omega)$, $\theta_{\vect{k}}$ is the angle between $\vect{k}(\omega)$ and the $k_z$-axis, the magenta arrows represent the group velocity $\vect{v}_g$ and the orange arrows the Poynting vector $\vect{S}$. Panel (d) shows a zoom into (a). In (d) the horizontal blue line represents $k_z=\omega/(0.5c)$ evaluated at energy 92.5\,meV. The contour plot (left panel) in (b) shows the momentum-resolved loss probability $P^{\text{bulk}}(k_{\bot};\omega)$ normalized to the maximum value (3\,a.u) for $v=0.1c$. The right panel in (b) shows the energy loss probability $\Gamma^{\text{bulk}}(\omega)$ obtained by integrating $P^{\text{bulk}}(k_{\bot};\omega)$ over $k_{\bot}$ up to $k^c_{\bot}=0.05\,\mathring{\text{A}}^{-1}$. (e) same as in (b) but considering $v=0.5c$. For this case the maximum of $P^{\text{bulk}}(k_{\bot};\omega)$ is 2.5\,a.u. Panels (c) and (f) depict the real part of the \textit{z}-component of the total electric field induced by the fast electron along the cylindrical coordinates $(R,z)$ for the energies: (c) 100\,meV (for $v=0.1c$)  and (f) 92.5, 100 and 105\,meV (for $v=0.5 c$). The field plots are normalized to the maximum value in each case: (c) $1\times10^{-6}\,\text{a.u.}$ and  (f) $7.5\times10^{-8}\,\text{a.u.}$. The insets in (c) and (f) illustrate the electron beam trajectory.}
\label{fig4}
\end{center}
\end{figure*}

To obtain spectroscopic information on the excitations in the lower Reststrahlen band, we calculate the EEL probability $\Gamma^{\text{bulk}}(\omega)$ by integration of $P^{\text{bulk}}(k_{\bot};\omega)$ in momentum space (right panel in Fig. \ref{fig4}(b)). Contrary to the upper Reststrahlen band, we observe a uniform and relatively small loss probability between $\text{TO}_{\parallel}$ and $\text{LO}_{\parallel}$ without the appearance of a sharp peak around $\text{LO}_{\parallel}$. We explain this finding by: 1) the large cutoff momenta ($\hbar k^c_{\bot}$) imposed by the aperture of the microscope detector and 2) the relationship between the energy and the transverse momentum of the polaritons in the lower band (see Fig. \ref{fig4}b left panel). Indeed, we observe in Fig. \ref{fig4}b that the asymptotic behavior of the blue dashed line tends to $\text{TO}_{\parallel}$ for large $k_{\bot}$. This shows that low energy hyperbolic phonon polaritons, close to $\text{TO}_{\parallel}$, largely contribute to the energy losses for large $k^c_{\bot}$ values. Contrary to the upper band, where the high momenta contribution to the electron energy losses comes from polaritons with high energy, close to $\text{LO}_{\bot}$ (Fig. \ref{fig3}b, left panel).  We address the reader to appendix \ref{appendC} where we show $\Gamma^{\text{bulk}}(\omega)$ in the lower Reststrahlen band for different cutoff values.

The excitation of phonon polaritons (within the lower Reststrahlen band) by the probing electron can be observed in Fig. \ref{fig4}c, where we show the real part of the \textit{z}-component of the total electric field induced at energy $\hbar\omega=100\,\text{meV}$. Analogously to the upper band, the oscillatory behavior of the field distribution along the \textit{z}-direction is governed by the transferred momentum $\hbar k_z$. Interestingly, the wake pattern is reversed compared to that for the upper Reststrahlen band (Figs. \ref{fig3}c and \ref{fig3}f), i.e., the wavefronts are propagating toward the electron beam \cite{Galyamin2011,Tao2016,Tao2019}. By plotting the group and phase velocity vectors onto the field plots (purple and black arrows, respectively; also plotted in Fig. \ref{fig4}a), we can clearly recognize that the projections of both vectors onto the radial axis (perpendicular to the electron beam trajectory) are antiparallel. This leads to a negative phase and positive group velocity relative to the Poynting vector direction (which points always away from the electron beam to preserve causality) along the radial axis. The negative phase velocity in the radial direction is a direct result of the phase velocity vector being nearly pependicular to the Poynting vector, both being rotated by $90^{\circ}$ as compared to the upper Reststrahlen band (where both phase and group velocities are positive relative to energy propagation in the radial direction, see Figs. \ref{fig3}c and \ref{fig3}f).

When the velocity of the electron is increased up to 50\,\% the speed of light, the $k_z$ component of the wavevector parallel to the beam trajectory is reduced. In this case, the matching between the red hyperbola and the horizontal blue line is prevented as observed in Fig. \ref{fig4}d. This mismatch of energy and momentum forbids the excitation of hyperbolic phonon polaritons. However, the blue line intersects the elliptical isofrequency surface of anisotropic bulk phonon polaritons (dielectric) above and below the lower Reststrahlen band (black and green dashed curves calculated for 105 and 92.5\,meV, respectively). The matching of energy and momentum at the intersections of the elliptical isofrequency surfaces leads to the excitation of the dielectric modes, as demonstrated by calculating the momentum-resolved loss probability $P^{\text{bulk}}(k_{\bot};\omega)$ (left panel of Fig. 4e). This loss probability is determined by the relationship between the energy of the elliptical polartions and the perpendicular momentum component (dashed blue lines, showing $\omega(k_{\bot})$ of the elliptical polaritons). The integration of $P^{\text{bulk}}(k_{\bot};\omega)$ in the reciprocal space subsequently yields small energy loss probabilities outside the Reststrahlen band, whereas inside the Reststrahlen band the loss probability is negligible due to absence of polariton excitations.

In Fig. \ref{fig4}f we show the total electric field induced by the electron beam for energies inside (marked 2) and outside (marked 1 and 3) the lower Reststrahlen band. We can observe the formation of wake patterns only for those energies where the dielectric modes are excited (marked as 1 and 3). Importantly, the wake wavefronts propagate outward the beam trajectory as a consequence of the group ($\vect{v}_g$) and phase velocities ($\vect{v}_p$) being parallel (positive) relative to the Poyting vector in the radial direction (Fig. \ref{fig4}d). We can also notice that the projection of these velocity vectors onto the \textit{z}-direction is positive. This demonstrates that the radial and \textit{z} projections of $\vect{v}_p$ and $\vect{v}_g$ for elliptical polaritons are positive, contrary to the hyperbolic regime (Reststrahlen bands) where one of the components is negative (Figs. \ref{fig3}c,f and \ref{fig4}c).
\subsection{Induced wake patterns and Cherenkov radiation}
We have shown in sections \ref{sec2}C (Figs. \ref{fig3}c,f) and  \ref{sec2}D (Figs. \ref{fig4}c,f,) that the field distributions produced by a fast electron traveling through h-BN can exhibit wake patterns. The excitation of these patterns (for energies inside and outside the Reststrahlen bands) is connected to the different mechanisms of energy losses experienced by the fast electron in the h-BN. In the following we discuss this connection.

First, it is worth noting that the excitation of the wake fields inside the Reststrahlen bands occurs for energies where electron losses appear (compare Fig. \ref{fig4}c with the image in Fig. \ref{fig4}f labeled as 2). As we pointed out, the electron energy losses within the Reststrahlen bands correspond to the excitation of hyperbolic phonon polaritons. This implies that the wake fields are associated to the excitation of coherent-charge density fluctuations\cite{Ritchie1974,Echenique1979,Echenique1979-2,DeAbajo1992,Liu2017,Tao2019} in the h-BN, namely, the phonon polaritons.

In contrast to the wake fields inside the Reststrahlen bands, the emergence of the wake patterns outside the bands (see Fig. \ref{fig4}, images labeled as 1 and 3) occurs due to a different physical process to that of the excitation of hyperbolic phonon polaritons. Outside the Reststrahlen bands the h-BN dielectric function is purely dielectric and thus the electron energy losses correspond to the radiation emitted by the electron when it passes through the medium with velocity larger than the speed of light (in the h-BN). This mechanism is known as Vavilov-Cherenkov radiation\cite{Cherenkov1934,Cherenkov1937,Tamm1937,Tamm1939,Lucas1970,Silcox1979,Ginzburg1996}. We have confirmed that the losses in this energy range are present even in the absence of damping in the material (not shown), confirming that the losses are due to Cherenkov radiation in this case. This only happens for velocities which fulfill
\begin{equation}
\label{2.o}
v>\frac{c}{\sqrt{\varepsilon_{\bot}}}.
\end{equation}
being consistent with the condition for excitation of Cherenkov radiation \cite{Rabernik2013}.

Finally, one can also note that the excitation of the wake fields in the lower Reststrahlen band depends on the electron velocity (compare Figs. \ref{fig4}c and \ref{fig4}f label 2). Indeed, for energies in the lower band one can deduce from Eq. \myref{2.j} that the wake patterns appear under the following condition:
\begin{equation}
\label{2.n}
\frac{\varepsilon_{\parallel}}{\varepsilon_{\bot}}<\frac{v^2}{c^2}\varepsilon_{\parallel} \hspace{2mm} \text{or} \hspace{2mm} v<\frac{c}{\sqrt{\varepsilon_{\bot}}},
\end{equation}
where only the real part of the dielectric function is considered. Interestingly, one can observe that the velocity of the fast electron fulfills different conditions for different energy ranges (compare Eqs. \myref{2.o} and \myref{2.n}). This difference is a direct consequence of the distinct physical processes in the excitation of the wake fields.

The different nature of the excitation of the wake fields outside and inside the Reststrahlen bands is also reflected in the angle $\theta_{\text{w}}=90^{\circ}-\theta_{\vect{k}}$ that the wake patterns sustain with respect to the electron beam trajectory. An analysis of this angle and its relationship with Eqs. \myref{2.o} and \myref{2.n} is developed at the end of appendix \ref{appendD}.

\subsection{Asymmetric wake patterns induced by tilting the electron beam trajectory}
As we pointed out in the last sections, the excitation of hyperbolic phonon polaritons can be controlled by the velocity of the fast electrons. In the following we study how steering of phonon polaritons can be controlled via the angle $\alpha$ between the electron beam trajectory and the h-BN optical axis.
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.70]{fig5.png}
\caption{Schematics of the electron traveling through the h-BN with velocity $\vect{v}=v(0,\sin\alpha,\cos\alpha)$ at an angle $\alpha$ with respect to the optical (\textit{z}-) axis of h-BN.}
\label{fig5}
\end{center}
\end{figure}

When the electron travels at an angle $\alpha$ relative to the h-BN optical axis (illustrated in Fig. \ref{fig5}), the condition for the conservation of energy and momentum given by Eq. \myref{2.g.1} ($\vect{k}\cdot\vect{v}=k_y v_y+ k_z v_z =\omega$) is represented by an inclined plane in momentum space (blue planes in Figs. \ref{fig6}a and \ref{fig6}d). The magnitude of the momentum transferred by the electron to the phonon polaritons (along the beam trajectory given by $\uvect{v}$) is still given by $\hbar k_{\uvect{v}}=\hbar \omega/v$. The polariton wavevector can be obtained from the intersection between the blue plane $\vect{k}\cdot\vect{v}=\omega$ and the isofrequency surfaces (red hyperboloids in Figs. \ref{fig6}a and \ref{fig6}d). Interestingly, we observe that the intersections are not cylindrically symmetric with respect to the \textit{z}-axis (Figs. \ref{fig6}a,c). This implies that the polaritonic wave will propagate asymmetrically with respect to the electron beam trajectory. Indeed, depending on the direction of propagation, the intersection between the blue planes and the red hyperboloids in Figs. \ref{fig6}a,d will occur at wavevectors $\vect{k}^{(1)}$ and $\vect{k}^{(2)}$ whose \textit{z}-component can be the same (symmetrical case) or different (asymmetrical case). To better understand the different  asymmetries in the propagation of the polaritonic wave we refer the reader to appendix \ref{appendD}, where we show the intersection of the blue plane and the red hyperboloids (Figs. \ref{fig6}a,d) for selected directions of the wavevector. Notice that the symmetric case is similar to the one we discussed in sections \ref{sec2}C and \ref{sec2}D. Therefore we will focus here on the analysis of the polariton propagation direction which shows the largest asymmetry, that is, the $k_yk_z$-plane.
\begin{figure*}[!ht]
\begin{center}
\includegraphics[scale=0.77]{fig6.png}
\caption{Isofrequency surfaces for (a) the upper and (d) the lower Reststrahlen bands for representative energies in each band: (a) 180\,meV and (d) 100\,meV. The blue inclined plane represents the condition for the conservation of energy and momentum in the non-recoil approximation: $\vect{k}\cdot\vect{v}=\omega$ for an electron with $v=0.1c$ and $\alpha=20^{\circ}$. Panels (b) and (e) next to each hyperboloid depict the intersection between the blue plane and the hyperboloids in the $k_yk_z$-plane. In these projections, the black arrows represent the two wavevector solutions $\vect{k}^{(1)}$, $\vect{k}^{(2)}$ with angles $\theta^{(1)}_{\vect{k}}$,  $\theta^{(2)}_{\vect{k}}$ with respect to the beam trajectory (blue dashed line), the magenta arrows represent the group velocities $\vect{v}^{(1)}_{g}$, $\vect{v}^{(2)}_{g}$ and the orange arrows the Poynting vectors $\vect{S}^{(1)}$, $\vect{S}^{(2)}$. The contour plots in (c) and (f) show the normalized real part of the \textit{z}-component of the total electric field in the $yz$-plane for the energies: (c) 180\,meV and (f) 100\,meV. We plot the field distributions for two different angles of the electron beam trajectory: $20^{\circ}$ (left panels) and $45^{\circ}$ (right panels). The maximum values of the field plots are: (c) $4\times 10^{-6}\,\text{a.u.}$ and (f) $>1.5\times 10^{-6}\,\text{a.u.}$}
\label{fig6}
\end{center}
\end{figure*}

We show in Fig. \ref{fig6}a the plane $\vect{k}\cdot\vect{v}=\omega$ for $v=0.1c$ (blue surface) and the isofrequency hyperboloid (red surface) for a representative energy in the upper Reststrahlen band ($\hbar\omega=180\,\text{meV}$). In Fig. \ref{fig6}b, we plot the projection of the intersection between the blue plane and the red hyperboloid in the $k_yk_z$-plane. The blue dashed line represents the electron beam trajectory and the black dashed line the $k_{z}$-axis. One can notice that the matching between the blue solid line and the red hyperbola (Fig. \ref{fig6}b) occurs at wavevectors $\vect{k}^{(1)}$ and $\vect{k}^{(2)}$ whose \textit{z}-component is different. Thus, the projections onto the \textit{z}-axis of the phase velocities $\vect{v}^{(1)}_{p}$ and $\vect{v}^{(2)}_{p}$ (parallel to $\vect{k}^{(1)}$ and $\vect{k}^{(2)}$, black arrows) are also different. Due to the hyperbolic shape of the isofrequency curve the \textit{z}-component of the group velocities $\vect{v}^{(1)}_{g}$ (parallel to the Poynting vector $\vect{S}^{(1)}$, right orange arrow in Fig. \ref{fig6}b) and $\vect{v}^{(2)}_{g}$ (parallel to the Poynting vector $\vect{S}^{(2)}$, left orange arrow in Fig. \ref{fig6}b) are also asymmetric. This difference (asymmetry) in the components of the two phase and group velocities leads to a highly asymmetric propagation of the polaritonic wave with respect to the electron beam trajectory.

The dependency of the polaritonic waves on the angle $\alpha$ can be observed in Fig. \ref{fig6}c, where we plot the real part of the \textit{z}-component of the total electric field produced by the fast electron at energy $\hbar\omega=180\,\text{meV}$ and $v=0.1c$ when $\alpha=20^{\circ}$ and $\alpha=45^{\circ}$. Similar to the parallel trajectory (sections \ref{sec2}C and \ref{sec2}D), one can notice the formation of wake patterns and the spatial periodicity of the field. This periodicity is determined by the momentum transferred along the beam trajectory ($\hbar k_{\uvect{v}}=\hbar \omega/v$) since the corresponding wavelength is $\lambda_{\uvect{v}}=2\pi/k_{\uvect{v}}$. Thus, the higher the energy of the polariton (the velocity of the electron) is, the smaller (bigger) $\lambda_{\uvect{v}}$ will be. The wake patterns formed by the field distribution are clearly asymmetric with respect to the beam trajectory. We observe (Fig. \ref{fig6}c) that the wake fields exhibit largest asymmetry as $\alpha$ is increased from $25^{\circ}$ (Fig. \ref{fig6}c, left panel) to $45^{\circ}$ (Fig. \ref{fig6}c, right panel). This is a direct consequence on how the electron transfers different momentum components, $\hbar k_y$ and $\hbar k_z$, to the polaritonic excitation (see Figs. \ref{fig6}b). One can notice from Fig. \ref{fig6}b that as $\alpha$ is increased, $k^{(1)}_z\approx 0$ and $\vect{k}^{(2)}$ tends to the asymptote of the red hyperbola. Therefore, for large angles $\alpha$ the polaritonic wave will propagate relative to the beam trajectory with a phase velocity close to zero on one side of the beam trajectory and with a constant phase velocity on the other side of the beam trajectory. These findings explain the absence of the wavefronts in Fig. \ref{fig6}c for $\alpha=45^{\circ}$ at the left side of the electron beam. It is worth noting that Fig. \ref{fig6}c corresponds to the propagation of the polaritonic wave in the $yz$-plane. However, for other propagation directions, the field distributions will be different.

In Figs. \ref{fig6}d-f we show the same analysis (electron beam trajectory tilted an angle $\alpha$ with respect to the h-BN optical axis) for a representative energy within the lower Reststrahlen band (100\,meV). Importantly, for this case the projections onto the \textit{y}-axis of the phase velocities $\vect{v}^{(1)}_{p}$ and $\vect{v}^{(2)}_{p}$ are antiparallel (negative) to the \textit{y}-component of the Poynting vectors $\vect{S}^{(1)}$ and $\vect{S}^{(2)}$. This yields an asymmetric wave propagating with negative phase velocity (Fig. \ref{fig6}f).

Additionally, the electron velocity $v$ allows to control the momentum transfer by the fast electron to the phonon polaritons (Eq. \myref{2.g.1}). Indeed, one can obtain the relationship between $v$ and the excitation of the asymmetric wake patterns by analyzing the wake angles $\theta^{(1)}_{\text{w}}=90^{\circ}-
\theta^{(1)}_{\vect{k}}$ and $\theta^{(2)}_{\text{w}}=90^{\circ}-
\theta^{(2)}_{\vect{k}}$ (Figs. \ref{fig6}c,f). We refer the reader to appendix \ref{appendD} where we derived this relationship. For completeness, we show in appendix \ref{appendE} the electron energy losses experienced by a fast electron traveling through h-BN in tilted trajectories with respect to the h-BN optical axis.

We have found that the excitation of the polaritonic wave is highly dependent on the orientation of the electron beam trajectory with respect to the h-BN crystallographic arrangement. Thus, while the speed of the electron serves as a means to excite the polaritonic wave or not, the orientation of the electron beam trajectory can serve to control the direction of the polaritonic excitation.

\section{Excitation of Dyakonov surface phonon polaritons in h-BN by a localized beam of fast electrons\label{sec3}}

We next study the EELS signal when the electron beam is traveling above an h-BN semi-infinite surface. The interface between vacuum and h-BN lies on the \textit{yz}-plane, as depicted in Fig. \ref{fig7}, with the \textit{y}-axis in the direction of $\varepsilon_{\bot}$ and the \textit{z}-axis in the direction of the h-BN optical axis. The electron travels in vacuum at a distance $x_0$ from the surface (we will refer to this distance as the impact parameter) with velocity $v$ parallel to the optical axis of the h-BN. A schematic representation of the probing electron-surface system is shown in Fig. \ref{fig7}.

Surfaces of uniaxial materials with optical axis parallel to the surface support a specific kind of surface waves, the so-called Dyakonov waves \cite{Dyakonov1988,Torner2008}. When either $\varepsilon_{\bot}$ or $\varepsilon_{\parallel}$ is negative (as in the case of the Reststrahlen bands in h-BN), surface polaritons called Dyankonov surface polaritons \cite{Talebi2019} can propagate along the surface. Recently, Dyakonov surface phonon polaritons have been observed by scattering-type scanning near-field optical microscopy (s-SNOM) at the edges of h-BN flakes \cite{Li2017,Hillenbrand2017} as well as by STEM-EELS \cite{Konecna2017,Talebi2016}. In the latter experiments, the probing electrons were passing outside the flake edge in a perpendicular trajectory. However, the excitation and detection of Dyakonov surface phonon polaritons with an electron beam parallel to an extended surface has not been described yet.

In the following, we first describe the Dyakonov surface phonon polariton modes that exist at the interface between h-BN and vacuum. We then show that a localized beam of fast electron can couple to these polaritons and we analyse the corresponding EEL spectra and their polariton wake patterns. Importantly, we find that surface Dyakonov phonon polaritons are excited only in the upper Reststrahlen band. Therefore, our analysis and calculations are restricted to this energy range.
\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.67]{fig7.png}
\caption{Schematics of the probing  electron traveling with velocity $v$ at a distance $x_0$ parallel to a h-BN surface. The optical axis of the h-BN crystal lattice is parallel to the h-BN surface. Label I refers to vacuum, while label II refers to h-BN.}
\label{fig7}
\end{center}
\end{figure}
\subsection{Surfaces modes in h-BN}

According to Dyakonov's theory \cite{Dyakonov1988}, the interface described in Fig. \ref{fig7} supports electromagnetic waves that propagate along it and their associated electromagnetic fields decay exponentially perpendicular to the interface \cite{Dyakonov1988,Torner2008,Cojocaru2014,Gonzalo2019}. These surface waves can be expressed as a linear superposition of the four following modes propagating along the interface: (i) a transverse electric (TE) mode, (ii) a transverse magnetic (TM) mode  [the corresponding  fields decay into the vacuum, upper half space in Fig. \ref{fig7} labeled I], (iii) an ordinary mode, and (iv) an extraordinary mode [the corresponding fields decay exponentially into h-BN, lower half space in Fig. \ref{fig7} labeled II]. Following this scheme the electric field in each media can be written as,
\begin{subequations}
\label{3.a}
\begin{align}
\vect{E}_{\text{I}}(x>0,y,z)&=(\vect{A}_{\text{TE}} + \vect{A}_{\text{TM}}) e^{-\kappa_{\text{I}} x} e^{i (k_y y +k_z z)},\\
\vect{E}_{\text{II}}(x<0,y,z)&=(\vect{A}_{\text{o}} e^{\kappa^{o}_{\text{II}} x} +  \vect{A}_{\text{e}} e^{\kappa^{e}_{\text{II}} x} )e^{i (k_y y +k_z z)},
\end{align}
\end{subequations}
where harmonic dependency in time has been assumed, and $\vect{A}_{\text{TE}},\vect{A}_{\text{TM}},\vect{A}_{\text{o}},\vect{A}_{\text{e}}$ are the amplitudes of each mode. The wavevector of each mode is given by
\begin{subequations}
\label{3.b}
\begin{align}
\vect{k}_d&=(i\kappa_{\text{I}},k_y,k_z) \hspace{2mm} \text{TE,\, TM},\\
\vect{k}_o&=(-i\kappa^{o}_{\text{II}},k_y,k_z) \hspace{2mm} \text{ordinary},\\
\vect{k}_e&=(-i\kappa^{e}_{\text{II}},k_y,k_z) \hspace{2mm} \text{extraordinary},
\end{align}
\end{subequations}
where $\kappa_{\text{I}},\kappa^{o}_{\text{II}},\kappa^{e}_{\text{II}}>0$ and $k_y,k_z \in \mathbb{C}$ need to fulfill the following conditions
\begin{subequations}
\label{3.c}
\begin{align}
\label{3.c.a}
&\kappa^2_{\text{I}}=k_y^2+k_z^2-(\omega/c)^2  \hspace{2mm} \text{vacuum},\\
\label{3.c.b}
&(\kappa^{o}_{\text{II}})^2=k_y^2+k_z^2- \varepsilon_{\bot} (\omega/c)^2 \hspace{2mm} \text{ordinary},\\
\label{3.c.c}
&(\kappa^{e}_{\text{II}})^2=k_y^2+\frac{\varepsilon_{\parallel}}{\varepsilon_{\bot}}k_z^2- \varepsilon_{\parallel} (\omega/c)^2 \hspace{2mm} \text{extraordinary}.
\end{align}
\end{subequations}
Applying boundary conditions imposed by Maxwell's equations at the interface between vacuum and h-BN, one obtains the following relationship \cite{Dyakonov1988,Zhu2016,Gonzalo2019}
\begin{equation}
\label{3.d}
(\kappa_{\text{I}}+ \kappa^{e}_{\text{II}})(\kappa_{\text{I}} + \kappa^{o}_{\text{II}})(\kappa_{\text{I}} + \varepsilon_{\bot} \kappa^{e}_{\text{II}}) =(\omega/c)^2 (\varepsilon_{\parallel}-1)(1-\varepsilon_{\bot})\kappa_{\text{I}},
\end{equation}
which together with the set of Eqs. \myref{3.c.a}-\myref{3.c.c} determines the in-plane wavevector $(k_y,k_z)$ of the Dyakonov waves.

It is worth noting that Dyakonov's original work \cite{Dyakonov1988} was derived for positive $\varepsilon_{\bot}$ and $\varepsilon_{\parallel}$. However Eq. \myref{3.d} is still valid when $\varepsilon_{\bot}<0$ and $\varepsilon_{\parallel}>0$ \cite{Cojocaru2014,Zhu2016}.  Since negative values in the real part of the dielectric components support the excitation of polaritonic states, Dyakonov surface waves sustained in h-BN in the mid-infrared region are thus called Dyakonov surface phonon polaritons.

In Fig. \ref{fig8} we plot the isofrequency contour (red curve) of the h-BN surface polariton for an energy within the upper Reststrahlen band (193\,meV), obtained from Eqs. \myref{3.c.a}-\myref{3.c.c} and \myref{3.d}. For comparison, we show a cut ($k_yk_z$-plane) of the isofrequency surface of the hyperbolic volume polariton (black dashed line obtained from Eq. \myref{2.c}). We find that the isofrequency curve of the surface polariton is a hyperbola, similar to that of the volume polariton particularly for small momenta. At large momenta, on the other hand, the opening angle of the isofrequency contour of the surface polariton, $\theta_{\text{s}}$, is smaller than that of the volume polariton $\theta_{\text{v}}$, demonstrating that the dispersion of Dyakonov phonon polaritons is different to the one obtained for the bulk hyperbolic phonon polaritons.
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.85]{fig8.png}
\caption{The red solid hyperbola represents the isofrequency curve obtained with Eqs. \myref{3.c.a}-\myref{3.c.c} and \myref{3.d} for the surface phonon polariton. While the black dashed hyperbola represents the isofrequency curve obtained using Eq. \myref{2.c} (setting $k_x=0$) for the bulk phonon polariton. Both curves are calculated for a representative energy in the upper Reststrahlen band, 193\,meV.}
\label{fig8}
\end{center}
\end{figure}

\subsection{Electron energy loss probability}

The excitation of Dyakonov surface phonon polaritons by fast electron beams can be revealed by electron energy loss spectra. In the following we describe the strategy to obtain the momentum-resolved loss probability, $P^{\text{surf}}(k_y;\omega)$, and the EEL probability, $\Gamma^{\text{surf}}(\omega)$, when the probing electron travels above the h-BN surface (see Fig. \ref{fig7}).

To calculate $\Gamma^{\text{surf}}(\omega)$, following Eq. \myref{2.f}, one needs to obtain the induced electric field, $\vect{E}^{\text{ind}}(\vect{r}_e;\omega)$, along the electron beam trajectory. To that end we obtain $\vect{E}^{\text{ind}}(\vect{r};\omega)$ by solving Maxwell's equations in the presence of vacuum-h-BN interface, assuming that the electron travels in vacuum with constant velocity $v$ and impact parameter $x_0$ (Fig. \ref{fig7}). Considering the boundary conditions at the interfaces ($x=0$), one finds the induced electric field in vacuum (region I in Fig. \ref{fig7}):
\begin{equation}
\label{3.e}
\vect{E}^{\text{ind}}_{\text{I}} (x,k_y,k_z;\omega)=(b_{\text{I}},d_{\text{I}},g_{\text{I}})\,\tilde{\rho}\,e^{-\kappa_{\text{I}} x},
\end{equation}
with $b_{\text{I}},d_{\text{I}},g_{\text{I}}$ being the coefficients involving the dielectric functions at both sides of the interface and $\tilde{\rho}=-2\pi e \delta(\omega-k_z v) e^{-\kappa_{\text{I}} x_0}/\varepsilon_0$.  We refer to appendices \ref{appendF} and \ref{appendG} for a detailed derivation of the total and induced electric fields at each half space (vacuum and h-BN).

By Fourier transforming $\vect{E}^{\text{ind}}_{\text{I}}(x,k_y,k_z;\omega) \mapsto \vect{E}^{\text{ind}}_{\text{I}}(\vect{r};\omega)$ in Eq. \myref{3.e} and inserting its value into Eq. \myref{2.f}, we find that $\Gamma^{\text{surf}}(\omega)$ can be written as
\begin{align}
\label{3.f}
\Gamma^{\text{surf}}(\omega)= \int_0^{k^c_y} \text{d}k_y \, P^{\text{surf}} (k_y;\omega),
\end{align}
where
\begin{equation}
\label{3.g}
P^{\text{surf}}(k_y;\omega) =  - \frac{e^2}{\pi^2 \epsilon_0 \hbar \omega v}   \text{Re} \left[g_{\text{I}} \,  e^{-2 \kappa_{\text{I}} x_0} \right]\Big\lvert_{k_z=\omega/v},
\end{equation}
is the probability that the electron transfers a transverse momentum $\hbar k_y$ (\textit{y}-component of the momentum) upon loosing energy $\hbar\omega$. Notice that the \textit{z}-component of the wavevector in $P^{\text{surf}}(k_y;\omega)$ is fixed by $k_z=\omega/v$, implying that the electron still transfers a parallel momentum equal to $\hbar \omega/v$.  The integration of Eq. \myref{3.f} is performed up to the  cutoff value $k^c_y$, which is determined by the aperture of the EELS detector.

As we discussed in section \ref{sec2}, the spectrum of the momentum-resolved loss probability and the EEL probability provides information on the properties of the excited modes in the anisotropic medium. We thus show in the following the relationship between these two quantities and the excitation of Dyakonov surface phonon polaritons.
\subsection{Excitation of surface phonon polaritons}
\begin{figure*}[!ht]
\begin{center}
\includegraphics[scale=0.83]{fig9.png}
\caption{The left panel in (a) displays the momentum-resolved loss probability $P^{\text{surf}}(k_y;\omega)$ normalized to the maximum value (3\,a.u.) in the vicinity of the upper Reststrahlen band for $x_0=10\,\text{nm}$ and $v=0.1c$. The right panel in (a) shows the EEL probability $\Gamma^{\text{surf}}(\omega)$ obtained by integrating $P^{\text{surf}}(k_y;\omega)$ over $k_y$ up to $k^c_{y}=0.09\,\mathring{\text{A}}^{-1}$. (c) same as in (a) but considering $v=0.5c$. For this case the maximum value of the momentum-resolved loss probability is 1\,a.u. The color maps in (b) and (d) show the real part of the \textit{z}-component of the induced electric field for the energies: 193 (marked 1, 3) and 198\,meV (marked 2, 4). The top panels in (b) and (d) correspond to the in-plane views ($yz$ plane) of the induced field, while the bottom panels correspond to the out-of-plane views ($xz$ plane).  The field plots are normalized with respect to the maximum value in each case. For the top panels: (b.1) $1\times10^{-4}\,\text{a.u.}$, (b.2) $7.5\times10^{-5}\,\text{a.u.}$, (d.3) $7.5\times10^{-6}\,\text{a.u.}\,$  and (d.4) $5\times10^{-6}\,\text{a.u.}$ For the bottom panels: (b.1) $4\times10^{-5}\,\text{a.u.}$, (b.2) $2\times10^{-5}\,\text{a.u.}$ and (d) $1.5\times10^{-6}\,\text{a.u.}$}
\label{fig9}
\end{center}
\end{figure*}

As pointed out above, the parallel momentum $\hbar k_z$ transferred by the fast electron to the phonon polaritons is determined by the relation $k_z=\omega/v$. Similarly to the bulk analysis of section \ref{sec2}, this relationship represents a horizontal line in the $k_yk_z$ representation of Fig. \ref{fig8}. Thus, the transferred momentum can be determined by the crossing between this horizontal line ($k_z=\omega/v$) and the isofrequency hyperbolas obtained from Eqs. \myref{3.c.a}-\myref{3.c.c} and \myref{3.d}. From the latter equations one can obtain the relationship between the \textit{y}-component of the polariton wavevector, $k_y$, and $\hbar\omega$, which is shown in the left panel of Fig. \ref{fig9}(a) (dashed blue curve). We also plot the momentum-resolved loss probability $P^{\text{surf}}(k_y;\omega)$ for energies around the upper Reststrahlen band. The probing electron is traveling above the h-BN surface with an impact parameter of 10\,nm and a velocity $v=0.1c$. Some similarities between $P^{\text{surf}}(k_y;\omega)$ and $P^{\text{bulk}}(k_{\bot};\omega)$ (Fig. \ref{fig3}b, left panel) become apparent. For instance, the highest values of $P^{\text{surf}}(k_y;\omega)$ (red and yellow colors in Fig. \ref{fig9}a) coincide perfectly with the blue dashed curve, demonstrating that the electron energy losses in the upper band are caused mainly due to the excitation of hyperbolic phonon polaritons. However, by comparing Figs. \ref{fig3}b and \ref{fig9}a we recognize that the asymptotic behavior (at large momenta) of $P^{\text{surf}}(k_y;\omega)$ occurs at a lower energy compared to the asymptotic behavior of $P^{\text{bulk}}(k_{\bot};\omega)$. While $P^{\text{bulk}}(k_{\bot};\omega)$ tends to the $\text{LO}_{\bot}$ phonon energy, $P^{\text{surf}}(k_y;\omega)$ tends to the surface optical ($\text{SO}_{\bot}$) phonon energy given by the condition $\varepsilon_{\bot}(\omega_{\text{SO}_{\bot}})=-1$ (derived from Eqs. \myref{3.c.a}-\myref{3.c.c}  and \myref{3.d} for large momenta). Importantly, the latter is a fingerprint of the excitation of surface polariton modes. In our case (electron traveling in vacuum above the h-BN surface) these surface modes correspond to Dyakonov surface phonon polaritons. We confirm this by integrating $P^{\text{surf}}(k_y;\omega)$ over $k_y$ up to a cutoff momentum $\hbar k^c_y$, which yields the EEL probability $\Gamma^{\text{surf}}(\omega)$ (right panel of Fig. \ref{fig9}b). A clear peak can be observed at the $\text{SO}_{\bot}$ phonon energy. This loss peak is slightly asymmetric with a broader tail for lower energies in the Reststrahlen band compared to that for larger energies in the band. Notice that for energies above $\text{SO}_{\bot}$ the loss spectrum displays a shoulder arising from background losses present in the entire upper band at small momentum (red blurred area for small momentum in the left panel of Fig. \ref{fig9}a).

The excitation of Dyakonov surface phonon polaritons (within the upper Reststrahlen band) by the probing electron can be observed in real space in Fig. \ref{fig9}b, where we show the real part of the \textit{z}-component of the induced electric field at energies 193\,meV (marked as 1) and 198\,meV (marked as 2). The top panels correspond to the evaluation of $\text{Re}(E^{\text{ind}}_z(\vect{r};\omega))$ in the $yz$-plane (in-plane at the interface), and the bottom panels to the evaluation in the $xz$-plane (lateral view, containing the electron trajectory). One can recognize from the in-plane views (Figs. \ref{fig9}b marked as 1) the formation of wake patterns and the oscillatory behavior of the induced field in the \textit{z}-direction. Similarly to the field distribution shown in Fig. \ref{fig3}c, the spatial periodicity along the \textit{z}-direction is connected with the parallel wavevector component $k_z=\omega/v$, since $\lambda_z=2\pi/k_z$. Moreover, the wake wavefronts show interesting propagation patterns both in the transverse direction from the beam trajectory as well as into the h-BN.

In the top panel of Fig. \ref{fig9}b (image labeled as 1), the wavefronts along the \textit{y}-direction propagate with positive phase and group velocities relative to the Poynting vector. Indeed, we find that the dashed blue curve in Fig. \ref{fig9}a has a positive slope ($\text{d}\omega/\text{d} k_y>0$), indicating that the projections onto the \textit{y}-axis of the group and phase velocities are parallel (positive). We also notice that Dyakonov surface phonon polaritons are confined to the interface with penetration of the field into the h-BN interface (Fig. \ref{fig9}b, bottom image labeled as 1). For energies larger than that of the $\text{SO}_{\bot}$ phonon, Dyakonov surface phonon polaritons are not excited (Fig. \ref{fig9}b, image labeled as 2). Thus, the induced field distributions for those energies correspond to the reflection of the electron electromagnetic field at the h-BN surface (Fig. \ref{fig9}b, top image labeled as 2). We can also notice that the field penetrates into the h-BN (bottom panel 2 of Fig. \ref{fig9}b), which is connected with the presence of the red blurred region corresponding to the losses appearing for lower momenta in Fig. \ref{fig9}a (left panel).

When the velocity of the probing electron is increased up to 50\,\% the speed of light, the momentum parallel to the beam trajectory, $\hbar k_z$, is reduced and so does the $k_y$ component of the Dyakonov surface phonon polariton. By calculating the momentum-resolved loss probability $P^{\text{surf}}(k_y;\omega)$ (left panel of Fig. \ref{fig9}c) and the EEL probability $\Gamma^{\text{surf}}(\omega)$ one obtains a similar behavior as in Fig. \ref{fig9}a for $v=0.1c$, except for a one order of magnitude reduction of both $k_y$ and the value of the loss probability.

The differences in the properties of the Dyakonov surface phonon polaritons launched by the fast electron beam can be observed in Fig. \ref{fig9}d, where we show the real part of the \textit{z}-component of the induced electric field for $v=0.5c$ at energies 193\,meV and 200\,meV. Notice that the spatial periodicity $\lambda_z$ of the polariton is longer in this situation compared to that in Fig. \ref{fig9}b as a result of the increased electron velocity. Also, the penetration of the field into the h-BN medium is larger compared to that  in Fig. \ref{fig9}b. This increase in the penetration depth can be attributed to the increase of the background losses present in the entire upper band (blurred red are in the left panel of Fig. \ref{fig9}c).

For completeness and similar to the analysis presented above, we study in appendix \ref{appendH} the excitation of phonon polaritons in the lower Reststrahlen band by a fast electron traveling in aloof trajectory. In the appendix we show the momentum-resolved loss probability ($P^{\text{surf}}(k_y;\omega)$), the EEL probability ($\Gamma^{\text{surf}}(\omega)$) and the wake patterns for this energy range.
\section{Remote excitation of bulk phonon polaritons\label{sec4}}
We have shown in Figs. \ref{fig9}b and \ref{fig9}d that the electric field penetrates into the bulk of the h-BN semi-infinite surface, which is surprising, as one does not expect the excitation of volume modes in isotropic materials for electron beam trajectories outside the material. By comparing the angles of the wake patterns, we demonstrate that indeed volume modes are excited in h-BN with external beam trajectories.
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.7]{fig10.png}
\caption{(a) Real part of the \textit{z}-component of the total electric field in the \textit{yz}-plane produced by a fast electron traveling through h-BN parallel to its optical axis. (c) shows $\text{Re}(E^{\text{tot}}_{z})$ evaluated in the \textit{xz}-plane and $\theta_{\text{w}}$ is the angle between the \textit{z}-axis and the wake patterns formed by the bulk polariton. (b) shows the real part of the \textit{z}-component of the induced electric field produced by a fast electron traveling in vacuum 10\,nm above a semi-infinite h-BN surface. (d) shows $\text{Re}(E^{\text{ind}}_{z})$ evaluated in the \textit{xz}-plane and $\theta_{\text{w}_{\text{s}}}$ is the angle between the \textit{z}-axis and the wake patterns formed by the Dyakonov surface phonon polariton. We used for the calculation of the fields an electron velocity equal to $v=0.5c$ at energy $\hbar\omega=193\,\text{meV}$. The field plots are normalized with respect to the maximum value in each case: (a) $5\times10^{-7}\,\text{a.u.}$, (b) $7.5\times10^{-6}\,\text{a.u.}$, (c) $5\times10^{-7}\,\text{a.u.}\,$  and (d) $1.5\times10^{-6}\,\text{a.u.}$ The insets above (a) and (b) illustrate the geometry under consideration for each case.}
\label{fig10}
\end{center}
\end{figure}

We first calculated the angle $\theta_{\text{w}}$ of the wake wavefronts produced by the fast electron traveling through bulk h-BN with $v=0.5c$ at $\hbar\omega=193\,\text{meV}$ (Figs. \ref{fig10}a and \ref{fig10}c), obtaining a value of $\theta_{\text{w}}=32.35^{\circ}$. We compare $\theta_{\text{w}}$ with the angles of the wake wavefronts produced by the fast electron traveling in an aloof trajectory 10\,nm above the h-BN surface (Fig. \ref{fig10}b and \ref{fig10}d). From this comparison we find that: (i) the angle $\theta_{\text{w}_{\text{s}}}=24.67^{\circ}$ of the wake pattern at the h-BN surface (Fig. \ref{fig10}b) is different from $\theta_{\text{w}}$, and (ii) the angle of the wake pattern excited inside the h-BN is the same as $\theta_{\text{w}}$ (Fig. \ref{fig10}d).  This implies that volume modes are excited by the fast electron traveling in trajectories outside the anisotropic medium.

Importantly, these findings open the possibility of remotely exciting volume phonon polaritons. In contrast to isotropic materials, where an aloof electron beam only couples to surface modes, for anisotropic materials the energy and momentum matching between the electron and the polaritons allows for launching of bulk excitations.

\section{Summary\label{sec5}}

We have thoroughly analyzed the excitation of optical phonon polaritons in hexagonal boron nitride by focused electron beams for two relevant situations: when the electron travels through the h-BN bulk and when it travels in vacuum above a semi-infinite h-BN surface. For the bulk situation, we have observed that the electron couples to volume phonon polaritons. We demonstrated that the excitation of these polaritonic modes is strongly dependent on the electron velocity and on the angle between the optical axis of h-BN and the trajectory of the electron beam.  Furthermore, we have shown that Dyakonov surface phonon polaritons can be excited by a fast electron traveling above the h-BN surface. Interestingly, aloof electron beams are capable of exciting volume polaritons in the h-BN.

By a detailed mode analysis, we showed that the electron beam transfers a specific momentum to the modes. This momentum transfer determines the properties of the excited phonon polaritons, and thus controls their phase and group velocities, as well as their propagation direction. Importantly, we found that the propagation of the polaritonic waves is highly asymmetric with respect to the electron beam trajectory when the trajectory sustains an angle relative to the h-BN optical axis.

Our findings may offer a way to steer and control the propagation of the polaritonic waves excited in hyperbolic materials. Although we studied the specific material h-BN, our findings can be generalized and can serve as an guide for the correct interpretation of the different excited modes and loss channels encountered in EELS experiments of uniaxial materials.

\section{Acknowledgements\label{sec6}}

C.M.-E. thanks \'{A}lvaro Nodar for his help in the parallelization of the programming codes. J. A. and R. H. acknowledges Grant No. IT1164-19 for research groups of the Basque University system from the Department of Education of the Basque Government. R.H. further acknowledges financial support from the Spanish Ministry of Science, Innovation and Universities (national project RTI2018-094830-B-100 and the project MDM-2016-0618 of the Marie de Maeztu Units of Excellence Program).

redpajama_set_name: RedPajamaArXiv


Why Professional Hood Cleaning is Beneficial One who owns his or her own restaurant certainly has a lot of advantages to enjoy, as restaurants are popular throughout the world with so many different people. One must never forget, however, that running a restaurant also entails a lot of duty and responsibility which should never be ignored. For example, people who own restaurants need to make sure that their kitchens are clean, as this is one of the things which is required by law and important for the health of both employees and customers. One will be glad to know, then, that he or she does not need to do all of this work alone, as there are professionals who can be hired, professionals who are trained and experienced in the job of hood cleaning and the cleaning of different areas of any kitchen. Finding professionals who offer hood cleaning services and kitchen cleaning, then, will assure you of so many benefits, and when you take this step and hire them, you will definitely be able to enjoy many rewards altogether. When one hires professionals to do hood cleaning services, he or she will be able to benefit, first of all, because professionals like these offer only the highest standards and quality. If you know about the cleanliness of your restaurant's kitchen, you might know that it is one of the most important things you should take care of, as without it, you can be sure that your business will fall into failure. One will be happy to know that professionals who offer cleaning services meet the highest standards when it comes to cleanliness, giving them the assurance of many benefits. Hiring professionals to do complex jobs like hood cleaning in your restaurant's kitchen is also wonderfully beneficial because when you take this step, you can be sure that everyone will be able to enjoy a higher level of health altogether. If you have ever tried cleaning your own kitchen, you might know that the job is complicated and difficult, and that there are many areas which are difficult to deal with – when you do not deal with these areas well, you can have problems like mold and other substances lingering there, which is certainly dangerous to anyone's health. One will be happy to know, then, that professionals who are certified in the job of kitchen cleaning can achieve a very thorough and careful cleanliness. In the long run, then, people can be assured that greater health can be enjoyed by everyone who works in the restaurant and everyone who eats its food.


text:
Current Topics in Medicinal Chemistry, Volume 7 - Number 8
Editorial [Hot Topic: Phospholipase Inhibition: Medicinal Chemistry and Therapeutic Potential of PLA2 Inhibitors (Guest Editor: Dr. K.S. Rangappa)]
K. S. Rangappa
Snake Venom Phospholipase A2 Inhibitors: Medicinal Chemistry and Therapeutic Potential
Silvana Marcussi, Carolina D. Sant'Ana, Clayton Z. Oliveira, Aristides Quintero Rueda, Danilo L. Menaldo, Rene O. Beleboni, Rodrigo G. Stabeli, Jose R. Giglio, Marcos R. M. Fontes and Andreimar M. Soares
Structural Elements of Ligand Recognition Site in Secretory Phospholipase A2 and Structure-Based Design of Specific Inhibitors
Nagendra Singh, Rishi K. Somvanshi, Sujata Sharma, Sharmistha Dey, Punit Kaur and Tej P. Singh
PLA2 Mediated Arachidonate Free Radicals: PLA2 Inhibition and Neutralization of Free Radicals by Anti-Oxidants – A New Role as Anti-Inflammatory Molecule
B. L. Nanda, A. Nataraju, R. Rajesh, K. S. Rangappa, M. A. Shekar and B. S. Vishwanath
Structural Biology of Recombinant Bovine Pancreatic Phospholipase A2 and its Inhibitor Complexes
K. Sekar
Chemistry and Structural Evaluation of Different Phospholipase A2 Inhibitors in Arachidonic Acid Pathway Mediated Inflammation and Snake Venom Toxicity
J. N. Narendra Sharath Chandra, K. C. Ponnappa, C. T. Sadashiva, B. S. Priya, B. L. Nanda, T. Veerabasappa Gowda, B. S. Vishwanath and K. S. Rangappa
Group IIA Secretory PLA2 Inhibition by Ursolic Acid: A Potent Anti-Inflammatory Molecule
A. Nataraj, C. D. Raghavendra Gowda, R. Rajesh and B. S. Vishwanath
Synthesis and Evaluation of Tricyclic Dipyrido Diazepinone Derivatives as Inhibitors of Secretory Phospholipase A2 with Anti-Inflammatory Activity
N. R. Thimmegowda, K. K. Dharmappa, C. S. Ananda kumar, M. P. Sadashiva, A. D. Sathish, B. L. Nanda, B. S. Vishwanath and K. S. Rangappa
, 7(8): 821
Kari Anne Myers

redpajama_set_name: RedPajamaCommonCrawl


A hospital visit doesn't just affect the patient. It affects the whole family — especially if you're traveling from far away. So it's important to plan ahead and to mentally prepare your child, yourself, and your family for the experience.


St. Joseph Regional Breast Care Center now has 3D mammography for the women in our communities.
The precise imaging produced by 3D mammography has definite potential for saving lives. It can find problem growths in the breast much earlier, at a stage when cancer, if found, is most successfully treated.
3D detects more cancers and it reduces the number of false-positive results given to patients. In most non-cancer cases 3D mammography by itself can ensure the patient is cancer-free, ruling out the need to be called back and wait for a biopsy.
It also offers clear imaging even in women, particularly young women, who have dense breast tissue, which can sometimes render conventional mammography ineffective.


text:
El mar Tirreno  es la parte del mar Mediterráneo que se extiende al oeste de la península itálica entre las islas de Córcega, Cerdeña y Sicilia y las costas continentales de Toscana, Lacio, Campania y Calabria.  Está unido al mar Jónico por el estrecho de Mesina y separado del mar de Liguria por la isla de Elba. Su profundidad máxima es de 3731 metros. Se encuentra próximo a la falla que divide África de Europa por lo que abundan las cadenas montañosas y los volcanes.

Toma el nombre del antiguo pueblo tirreno, más conocido como etrusco; Heródoto cuenta que, desde Anatolia occidental, el pueblo lidio emigró buscando una nueva patria guiado por el príncipe Tirreno, y que al llegar a las costas de la península itálica, en agradecimiento a su príncipe, tomaron el nombre de tirrenos.

Delimitación de la IHO
La máxima autoridad internacional en materia de delimitación de mares, la Organización Hidrográfica Internacional, considera el mar Tirreno como una subdivisión del mar Mediterráneo. En su publicación de referencia mundial, Limits of oceans and seas ('Límites de océanos y mares', 3.ª edición de 1953), le asigna el número de identificación 28 (e) y define sus límites de la forma siguiente:

Salidas

Hay cuatro salidas del Mar Tirreno (de norte a sur):

Cuencas

La cuenca del Tirreno se divide en dos cuencas (o llanuras), la llanura de Vavilov y la llanura de Marsili. Están separadas por la cresta submarina conocida como puente de Issel, en honor al geólogo italiano Arturo Issel.

Geología

El mar Tirreno es una cuenca de retroarco que se formó debido al retroceso de la placa de Calabria hacia el sureste durante el Neógeno. Los episodios de retroceso rápido y lento de la fosa formaron primero la cuenca de Vavilov y, después, la de Marsili. Los volcanes submarinos y el volcán activo Estrómboli se formaron porque el retroceso de la fosa produce una extensión en la placa de sobrecarga que permite que el manto se eleve por debajo de la superficie y se funda parcialmente. El magmatismo aquí también se ve afectado por los fluidos liberados de la losa.

Origen del nombre

Su nombre deriva del nombre griego de los etruscos, de los que los griegos decían que eran emigrantes de Lidia y estaban dirigidos por el príncipe Tirreno. Los etruscos se asentaron a lo largo de la costa de la actual Toscana y se referían a estas aguas como el "Mar de los Etruscos".

Véase también
 Lista de islas en el Adriático
 Mar Jónico
 Mar Adriático
 Mar de Liguria

Notas

redpajama_set_name: RedPajamaWikipedia


'Barbecue Pork' is an all time favorite for housewives. Supermarkets sell cooked barbecue pork and it comes in handy for dishes which need a slight flavor such to name a few – sauteed vegetables, fried rice, etc. etc.
If you are a lover of dim sum, 'Chimaki' is that sticky rice wrapped in bamboo shoots which with just a slight effort and preparation can be made at home.
The RED beet color is sort of addictive to me…. I've been trying to get the color to remain in a cake but with no luck….
Chinese Herbs Cooking (which we call it 'Yakuzen' in Japanese) has always facinated me whilst working in China – way back in year 2002, just that moment of the SARS outbreak.


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Companies Vs. Climate Change | The Bugs Are Coming!
That could encourage farmers to use more pesticides, which could cause further environmental harm, scientists said.
For every degree Celsius (two degrees Fahrenheit) that temperatures rise above the global historical average, the amount of wheat, corn, and rice lost to insects will increase by 10 to 25 percent, the study says. Temperate agricultural regions, like those in the United States and Western Europe, would be particularly hard hit.
The international Paris Agreement is designed to keep warming below two degrees Celsius, but the world's countries are far off track from meeting that goal.
Higher temperatures speed many insects' metabolisms, making them eat more. Their life cycles also get faster, causing them to reproduce more quickly. Both effects would diminish crop yields even as the human population continues to increase, putting additional strains on the global food supply, the study says.
To arrive at their estimates, Dr. Deutsch and his colleagues used statistical models to simulate the effects of global warming on insect feeding and reproduction. They focused on wheat, corn and rice crops because they account for 42 percent of the calories directly consumed by humans.
Other factors could help mitigate crop losses. Beneficial insects could also thrive in a warmer climate, said Michael Hoffmann, a professor of entomology and executive director of the Cornell Institute for Climate Smart Solutions, who was not involved in the study. Those insects could end up "offering some suppression of the pests, so that the damage may not be as great as they are suggesting," Dr. Hoffmann said.
Still, higher temperatures can spell bad news for thirsty crops regardless of insect activity. A study last year in the journal Nature Communications found that the pressures from increased summer temperatures could lead to a significant decline in agricultural yields. This summer's European heat wave, which is in keeping with patterns of climate change, reduced Germany's grain production by roughly 20 percent.
That study found that improved irrigation could offset at least some of the losses. But it is less clear if insecticides could help stave off multiplying pests.
"The one out of eight loaves of bread that we currently lose already reflects what we can do to manage crop losses to insects," said Dr. Deutsch. Pesticides could help where they're not already in use, but in other regions, "there's a real question as to whether or not they're already at their maximum effectiveness," he said.
In addition, pesticides can unintentionally harm other organisms, and some have been linked to human health problems. Their manufacture, transport and use also contribute to global warming.


When something goes wrong at the workplace, it could be for any number of reasons - power failure, system crashes, or even malicious software. But, before you go and assume that the problem is irreparable, check your cables for any abnormalities. Even the slightest cabling issue can cause big problems for your company's entire network.


text:
Voters in 12 parliamentary constituencies in Kelantan, Kedah and Perlis which I visited after the signing of the "Citizen's Declaration" of March 4 support Save Malaysia campaign calling for Najib's removal as well as democratic and institutional reforms
Kuala Perlis is my seventh stop in the visit to four parliamentary constituencies in Kedah and three parliamentary constituencies in Perlis yesterday and today.
Kangar (which includes Kuala Perlis) is the 92nd parliamentary constituency I am visiting since my six-month suspension from Parliament on Oct. 22 for demanding full and satisfactory accounting from the Prime Minister, Datuk Seri Najib for his RM55 billion 1MDB and RM2.6 billion "donation" twin mega scandals.
The seven parliamentary constituencies in Kedah and Perlis I visited in the last two days are: Sungai Patani, Merbau, Kuala Kedah, Alor Setar, Padang Besar, Arau and Kangar.
The purpose on my nation-wide visit of the parliamentary constituencies is to take the pulse of the people on Najib's twin mega scandals, and there is no doubt that uppermost in everyone's minds are teeming questions as where the billions in Najib's personal banking accounts came from, where they have gone to, and whether the RM2.6 billion (mushroomed now to RM4.2 billion) donation scandal is part of the RM55 billion 1MDB scandal.
So many question, and which are increasing by the day, which remain unanswered – sending out the disturbing message not only to Malaysians but to the world that Najib has got many things to hide!
In the whole process for the past year, Malaysia is acquiring the notoriety of being among the most corrupt nations in the world – how sad for patriotic Malaysians!
Since March 4, there is an added issue which I am taking the pulse of the people during my visit to the parliamentary constituencies - the people's reactions and feedback to the Citizen's Declaration to Save Malaysia signed by 45 political and civil society leaders, involving former Prime Minister, Tun Dr. Mahathir, former Deputy Prime Minister, Tan Sri Muhyiddin Yassin as well as former Cabinet Ministers, bridging the political divide with signatories from the government coalition and the Opposition, calling for Najib's removal as well as for democratic and institutional reforms.
I have visited five parliamentary constituencies in Kelantan (Ketereh, Machang, Tanah Merah, Tumpat and Kota Bahru), four parliamentary constituencies in Kedah (Sungai Patani, Merbok, Kuala Kedah, Alor Setar) and three parliamentary constituencies in Perlis (Padang Besar, Arau and Kangar) since March 4 to get the people's feedback to the Citizen's Declaration to Save Malaysia launched on March 4.
I find the voters in the 12 parliamentary constituencies in Kelantan, Kedah and Perlis which I visited after the signing of the "Citizen's Declaration" of March 4 support the Save Malaysia campaign calling for Najib's removal as well as for democratic and institutional reforms although they have some questions and reservations.
In the final analysis, however, they are in support of the larger picture and objective –that the paramount question is how to cure Malaysia of the great sickness which has caused the nation to fail in achieving the great national promises when we aspired when the nation attained Merdeka 59 years ago with our rich human talents and natural resources.
Why is Malaysia regressing in every field of human endeavor, whether political, economic, educational, human rights, the rule of law, national unity, and even in the field of sports – as the film "Ola Bola" remind Malaysians how we have fallen from a great Asian football nation in the early decades of independent nationhood to No. 171 in the current FIFA ranking for Malaysia!
Malaysia must shake off the spectre of Najib's twin mega scandals, which had been haunting and hounding Malaysia's national and international image for the past year.
The arrest of two Australian journalists in Kuching who wanted to question Najib about his twin mega scandals have again catapulted Malaysia into new international notoriety.
Malaysia has not recovered from the adverse international publicity over the recent ban on The Malaysian Insider and other repressive actions to continue to hide the truth about Najib's twin mega scandals from the Malaysian public, we are already landed with another "black eye" to the nation's frail and vulnerable international reputation.
Najib as Prime Minister should lay down the policy for the Attorney-General to follow and not the other way round – and he should make it clear to his chief law officer to be "hands off" the two Australian journalists to allow them to return to Australia without any further restraint as well as to revoke the ban on The Malaysian Insider and cease and desist from embarking on a spree to put more draconian laws on the statute books in the current session of Parliament.
Speech by Lim Kit Siang in Perlis DAP "Mana RM2.6 billion?" kopitiam ceramah held at Kuala Perlis, Perlis on Monday, 14th March 2016


Back in October and November when we were putting up the hoop house I wasn't sure what that really meant. Sure, I knew it would extend my growing season. I knew that it was more likely to be successful in allowing me to begin next year's season early rather than extending last year's growing season late.
But this morning I went down and lo, nearly all my spinach planted last week is sprouting with terrific germination (the best I've ever had from spinach).
We eat a lot of salad, but I think in the past 8 months or so I have bought one bag of spinach…maybe two.
I've also got healthy rosemary, kohlrabi, dill, and turnips. The perpetual spinach chard planted last spring is still going gangbusters.
I think I have watered twice – if that – since November.
That hoop house just might be a good investment after all.
This entry was posted in General on January 4, 2012 by mbonebrake.


text:
MAIAH MANSER
written by the partae May 29, 2017
What is your name and role within the band?
Maiah Manser. Singer/songwriter/composer/knit picker.
I started singing and performing when I was 4, but I didn't start singing in bands until I was 19 which were primarily jazz and soul.
I am now based in Los Angeles, but spent the last 7 years honing my craft and developing in Seattle.
Please give an example of your music writing process?
One of my songs, "Sweet Hell" came to me in an odd dream of a beautiful naked woman singing the melody directly to me. I woke with a start at 3am and went out and recorded the melody into my phone. The next day, I wrote the entire song. (haha) That's an unusual method though. I usually try to keep it fresh and new each time that I write, but what I've recently been doing is finding a chord progression that I love and discovering what melody hooks me. From there I develop words to a chorus, which feels like. I make sure to write very bluntly how I am feeling without pause and in my doing so, I haven't been able to really hate any of my songs since. I had an ex tell me a few years ago that he thought I wasn't saying what I really meant. Maybe one of the only profitable pieces that came out of that relationship (haha), but it has stuck with me to this day and changed my songwriting process forever.
I'm working on playing more shows and already started working on producing new songs about 6 months ago. Finding the right production takes me awhile, but it is the greatest gift when you can create the right moment.
What is your gear setup?
Playing solo, it's just me, a keyboard and an electro harmonix 45000 vocal loop pedal. When I have the band, it's me, my loop pedal, an analog synth keyboardist, and a drummer equipped with a electronic drum pad.
What do you like to do outside of music and does it affect your music?
I love to do visual art and dance. I don't have a lot of time for it these days, but I try to squeeze it in whenever I can. Growing up, visual art was close in line with music, but now I've found solace in drawing and painting. When I get overwhelmed, I can find peace in a pencil and a pad of paper or a contemporary dance class and I'm very thankful for that.
How would you describe your music genre?
My genre has been described as pop noir, but recently, for maybe the first time in my songwriting career I've been able to write happier sounding songs. It's amazing how environment can affect, change and develop your work.
Do you know any music theory?
I'm a complete music theory nerd. I went to Cornish College of the Arts for a couple years and studied in the jazz program. Knowing theory at first made it feel like a part of the magic was taken away, but now I fully embrace it and it's given me the opportunity to understand many more instruments across the board.
To keep being able to make music and having so much fun going in a little more of a pop route!
How did you get into music?
It's always been impossible for me to take naps, so when I was 4, during nap time I would sometimes terrorize the other kids around me, trying to keep them awake. Somehow the teachers discovered I could sing and would take me around to other classes and I would perform during nap time instead. When I was 6, I won a talent show singing "Edelweiss" and then I knew that music was what I wanted to do. No one in my family was musical, except for my grandmother and mother singing me to sleep, so I got a lot of ridicule and disbelief from my family on what I wanted to do. I remember when I was 8 that I started telling them: I'm going to be a professional harmony vocalist for a pop star, learn as much as I can and then through that, I'm going to develop my own career. I did just that.
What are you listening to at the moment?
Basically only Kendrick Lamar.
Who are your top 5 influences and icons?
Heavy question. Fiona Apple, TLC, The Little Mermaid, Nina Simone and Celine Dion.
When are you playing next?
I'm playing NYC on June 1st at Jan Hus Presbyterian Church and again in NYC on June 12th at Berlin.
www.maiahmanser.com
www.instagram.com/maiahmanser
www.facebook.com/maiahmansermusic
NEU YEUTH
UNCLE SCOTCHY OMB
JERRY WILLIAMS
Good Doogs
Kawehi
Safe To Swim
Leesta Vall Sound Recordings
BVRGER
Justin Erougian – Airwolf Paradise
Miillie Mesh
Tham


text:
Q: FFMPEG DirectShow: Could not run filter I'm trying to take a picture from my webcam using FFMpeg.
This is what I'm trying to run:
@echo off

:: Get date and time
for /F "usebackq tokens=1,2 delims==" %%i in (`wmic os get LocalDateTime /VALUE 2^>NUL`) do if '.%%i.'=='.LocalDateTime.' set ldt=%%j
set datetime=%ldt:~0,4%_%ldt:~4,2%_%ldt:~6,2%_%ldt:~8,2%_%ldt:~10,2%_%ldt:~12,2%

:: Capture snapshot through DirectShow using FFmpeg and save to disk.
ffmpeg.exe -f dshow -i video="1.3M WebCam" -vframes 1 D:\snapshot_%datetime%.jpg

Output:
    ffmpeg version N-63746-gfbaf73a Copyright (c) 2000-2014 the FFmpeg developers
    built on Jun  3 2014 22:10:20 with gcc 4.8.2 (GCC)
    configuration: --enable-gpl --enable-version3 --disable-w32threads --enable-av
    isynth --enable-bzlib --enable-fontconfig --enable-frei0r --enable-gnutls --enab
    le-iconv --enable-libass --enable-libbluray --enable-libcaca --enable-libfreetyp
    e --enable-libgme --enable-libgsm --enable-libilbc --enable-libmodplug --enable-
    libmp3lame --enable-libopencore-amrnb --enable-libopencore-amrwb --enable-libope
    njpeg --enable-libopus --enable-librtmp --enable-libschroedinger --enable-libsox
    r --enable-libspeex --enable-libtheora --enable-libtwolame --enable-libvidstab -
    -enable-libvo-aacenc --enable-libvo-amrwbenc --enable-libvorbis --enable-libvpx
    --enable-libwavpack --enable-libwebp --enable-libx264 --enable-libx265 --enable-
    libxavs --enable-libxvid --enable-decklink --enable-zlib
      libavutil      52. 89.100 / 52. 89.100
      libavcodec     55. 66.100 / 55. 66.100
      libavformat    55. 42.100 / 55. 42.100
      libavdevice    55. 13.101 / 55. 13.101
      libavfilter     4.  5.100 /  4.  5.100
      libswscale      2.  6.100 /  2.  6.100
      libswresample   0. 19.100 /  0. 19.100
      libpostproc    52.  3.100 / 52.  3.100
    [dshow @ 0000000002ac0480] Could not run filter
    video=1.3M WebCam: Input/output error

Can't figure out where the problem is.
When I run ffmpeg -list_devices true -f dshow -i dummy, I get:
ffmpeg version N-63746-gfbaf73a Copyright (c) 2000-2014 the FFmpeg developers
  built on Jun  3 2014 22:10:20 with gcc 4.8.2 (GCC)
  configuration: --enable-gpl --enable-version3 --disable-w32threads --enable-av
isynth --enable-bzlib --enable-fontconfig --enable-frei0r --enable-gnutls --enab
le-iconv --enable-libass --enable-libbluray --enable-libcaca --enable-libfreetyp
e --enable-libgme --enable-libgsm --enable-libilbc --enable-libmodplug --enable-
libmp3lame --enable-libopencore-amrnb --enable-libopencore-amrwb --enable-libope
njpeg --enable-libopus --enable-librtmp --enable-libschroedinger --enable-libsox
r --enable-libspeex --enable-libtheora --enable-libtwolame --enable-libvidstab -
-enable-libvo-aacenc --enable-libvo-amrwbenc --enable-libvorbis --enable-libvpx
--enable-libwavpack --enable-libwebp --enable-libx264 --enable-libx265 --enable-
libxavs --enable-libxvid --enable-decklink --enable-zlib
  libavutil      52. 89.100 / 52. 89.100
  libavcodec     55. 66.100 / 55. 66.100
  libavformat    55. 42.100 / 55. 42.100
  libavdevice    55. 13.101 / 55. 13.101
  libavfilter     4.  5.100 /  4.  5.100
  libswscale      2.  6.100 /  2.  6.100
  libswresample   0. 19.100 /  0. 19.100
  libpostproc    52.  3.100 / 52.  3.100
[dshow @ 00000000003103e0] DirectShow video devices
[dshow @ 00000000003103e0]  "1.3M WebCam"
[dshow @ 00000000003103e0] DirectShow audio devices
[dshow @ 00000000003103e0]  "Microphone (Realtek High Defini"
dummy: Immediate exit requested

redpajama_set_name: RedPajamaStackExchange


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