# Vavilov-Čerenkov radiation for parallel motion in three-dimensional
topological insulators

O. J. Franca<EMAIL_ADDRESS>Theoretische Physik III, Universität
Kassel, Heinrich-Plett-Straße 40, 34132 Kassel, Germany  Stefan Yoshi Buhmann
<EMAIL_ADDRESS>Theoretische Physik III, Universität Kassel,
Heinrich-Plett-Straße 40, 34132 Kassel, Germany

###### Abstract

Our study delves into the modifications observed in Vavilov-Čerenkov radiation
when its generating charged particle moves parallel to an interface formed by
two generic magnetoelectric media, focusing on topological insulators. We
compute the electromagnetic field through the Green’s function. Applying the
far-field approximation and the steepest descent method, we derive analytical
expressions for the electric field, revealing contributions from spherical and
lateral waves with topological origins. Subsequently, we analyze the angular
distribution of the radiation, particularly focusing on parallel motions in
close proximity to the interface. Our findings indicate that the radiation
along the Vavilov-Čerenkov cone is inhomogeneous and asymmetric. We analyze
the radiated energy at both sides of the interface. Finally, we discuss the
particle’s retarding force, which is notably enhanced in the ultrarelativistic
regime. We illustrate these results for the topological insulator TlBiSe2 and
the magnetoelectric TbPO4.

##  I Introduction

Over sixteen years have elapsed since the theoretical prediction TIs
Prediction 1 ; TIs Prediction 2  and experimental validation of the existence
Hsieh  of three-dimensional topological insulators (TIs), catalyzing the
research into these materials leading to the detection of new varieties of TIs
TIs variety . Three-dimensional TIs constitute an interesting new state of
matter exhibiting conducting surface states arising out of an unusual band
structure that belongs to a different topological class different from that of
ordinary semiconductors Anirban , being protected by time-reversal symmetry
and insulating bulk Hasan ; Qi Review . Originally, two types of TIs were
established: weak three-dimensional TIs with an even number of surface states
on all but a single surface, whereas strong ones exhibit an odd number of them
on each surface TIs Prediction 1 ; TIs Prediction 2 ; Morgenstern review . The
introduction of time-reversal symmetry breaking at the interface between a
conventional insulator and a 3D strong TI—accomplished through methods such as
magnetic coating or doping the TI with transition metal elements—results in
the opening of a gap in the surface states. This leads to a plethora of
phenomena, including the quantum anomalous Hall effect QAH ; Mogi , the
quantized magneto-optical effect Wu ; Dziom , the topological magnetoelectric
effect Hasan ; Qi PRB ; Okada , and the still to-be verified image magnetic
monopole effect Qi Science .

Despite the theoretical prediction of radiative effects in TIs OJF-LFU-ORT ;
OJF-SYB ; OJF-LFU , their applications remain scarcely studied. Among the
various types of radiation, the Vavilov-Čerenkov (VC) one has become an
important tool in high-energy particle physics, high-power microwave sources
and nuclear and cosmic-ray physics Jelley ; Jelley1  from the theoretical and
phenomenological side. VC radiation arises when charged particles traveling
through a dielectric medium with velocity $v$ higher than
$c/\sqrt{\varepsilon\mu}$ cherenkov ; Vavilov . Here $c$ is the speed of light
in vacuum, $\varepsilon$ denotes the permittivity of the medium, $\mu$ labels
its permeability, and $n=\sqrt{\varepsilon\mu}$ stands for the refraction
index of the material. Frank and Tamm were the first to explain this radiation
theoretically and revealed its unique polarization and directional properties
by means of classical electrodynamics FT . In transparent dielectric media, VC
radiation behaves as an extended mode and lies in a forward cone defined by
the angle $\theta^{C}=\cos^{-1}[c/vn]<\pi/2$ with respect to the direction of
the generating charge. Multiple applications of VC radiation in nuclear and
high-energy physics have been found for designing detectors Ypsilantis ; PADG
and particle discoveries Chamberlain ; Aubert . VC radiation has also been
actively studied beyond high-energy physics in other branches of physics
ranging from metamaterials Tao , magnetoelectric media ODELL ; OJF-LFU-ORT ,
chiral media Kolomenskii ; Galyamin 1 . Even medical applications like the
Čerenkov luminescence imaging Spinelli  and the interplay between surface
waves and VC radiation have recently been explored Hu-Lin-Wong .

As predicted by Veselago it is possible to reverse the direction of VC
radiation in left-handed materials Veselago  or, as the article OJF-LFU-ORT
theorized, for three-dimensional TIs, which are right-handed materials. In the
latter work, it was found that reversed VC radiation occurs in this class of
materials when its generating particle travels with a trajectory perpendicular
to the surface of the interface between the material and vacuum.

Based on these ongoing developments, this paper will be devoted to 3D TIs and
VC radiation but with a slightly different configuration. The aim of the
present work is to examine how VC radiation interacts with the interface
between two arbitrary strong 3D TIs with different permeabilities and
permeabilities when the charged particle moves parallel to this interface.
Henceforth, we will refer to this configuration as parallel Vavilov-Čerenkov
radiation to distinguish it from the perpendicular motion studied in Ref. OJF-
LFU-ORT . We will compare with the results of Bolotovskii Bolotovskii  for an
interface constituted by two ordinary media.

This paper is organized as follows. In Sec. II, we describe the modified
Maxwell equations which describe the electromagnetic response of three-
dimensional TIs from an effective field theory point of view. Also in this
section, we review the Green’s function method to obtain the time-dependent
electromagnetic field. Section III is devoted to analyzing the electromagnetic
field associated with parallel VC radiation. In Sec. IV, we present angular
distributions of the parallel VC radiation for the upper and lower hemispheres
corresponding to the reflected and transmitted radiation respectively. Section
V is devoted to obtaining an analytical expression for the radiated energy at
both sides of the interface. In Section VI we study the retarding force
experienced by the particle. Finally, a summary and concluding remarks of our
results can be found in Section VII.

##  II Modified Maxwell Equations

The Maxwell equations and constitutive relations of the involved material
constitute our natural starting point. For a three-dimensional TI, these are
Qi PRB ; Chang-Yang ; Obukhov-Hehl

$\displaystyle\nabla\cdot\mathbf{B}\left(\mathbf{r};\omega\right)$
$\displaystyle=$ $\displaystyle 0\;,$ (1)
$\displaystyle\nabla\times\mathbf{E}\left(\mathbf{r};\omega\right)$
$\displaystyle=$
$\displaystyle\mathrm{i}\omega\mathbf{B}\left(\mathbf{r};\omega\right)\;,$ (2)
$\displaystyle\nabla\cdot\mathbf{D}\left(\mathbf{r};\omega\right)$
$\displaystyle=$ $\displaystyle\varrho\left(\mathbf{r};\omega\right)\;,$ (3)
$\displaystyle\nabla\times\mathbf{H}\left(\mathbf{r};\omega\right)+\mathrm{i}\omega\mathbf{D}\left(\mathbf{r};\omega\right)$
$\displaystyle=$ $\displaystyle\mathbf{j}\left(\mathbf{r};\omega\right)\;,$
(4)

where $\varrho(\mathbf{r};\omega)$ and $\mathbf{j}(\mathbf{r};\omega)$ denote
the source term for electromagnetic waves generated by external charges and
currents, respectively, $\mathbf{E}\left(\mathbf{r};\omega\right)$ stands for
the electric field and $\mathbf{B}\left(\mathbf{r};\omega\right)$ is the
induction field. For TIs, both fields are related with the displacement field
$\mathbf{D}\left(\mathbf{r};\omega\right)$ and the magnetic field
$\mathbf{H}\left(\mathbf{r};\omega\right)$ via

$\displaystyle\mathbf{D}\left(\mathbf{r};\omega\right)$ $\displaystyle=$
$\displaystyle\varepsilon_{0}\varepsilon\left(\mathbf{r};\omega\right)\mathbf{E}\left(\mathbf{r};\omega\right)+\frac{\alpha\Theta\left(\mathbf{r};\omega\right)}{\pi\mu_{0}c}\mathbf{B}\left(\mathbf{r};\omega\right)$
(5) $\displaystyle+\mathbf{P}_{N}\left(\mathbf{r};\omega\right)\;,$
$\displaystyle\mathbf{H}\left(\mathbf{r};\omega\right)$ $\displaystyle=$
$\displaystyle\frac{\mathbf{B}\left(\mathbf{r};\omega\right)}{\mu_{0}\mu\left(\mathbf{r};\omega\right)}-\frac{\alpha\Theta\left(\mathbf{r};\omega\right)}{\pi\mu_{0}c}\mathbf{E}\left(\mathbf{r};\omega\right)$
(6) $\displaystyle-\mathbf{M}_{N}\left(\mathbf{r};\omega\right)\;,$

where $\alpha$ is the fine structure constant and
$\varepsilon\left(\mathbf{r};\omega\right)$,
$\mu\left(\mathbf{r};\omega\right)$, and
$\Theta\left(\mathbf{r};\omega\right)$ are the dielectric permittivity,
magnetic permeability, and axion coupling, respectively. Here the
$\mathbf{P}_{N}\left(\mathbf{r};\omega\right)$ and
$\mathbf{M}_{N}\left(\mathbf{r};\omega\right)$ terms are the noise
polarization and magnetization, respectively. These are Langevin noise terms
that represent absorption within the material Scheel-Buhmann .

For weak three-dimensional TIs the axion coupling
$\Theta\left(\mathbf{r};\omega\right)$ takes even multiples of $\pi$, whereas
for strong three-dimensional TIs, it takes odd multiples of $\pi$, with the
magnitude and sign of the multiple given by the strength and direction of the
time-symmetry-breaking perturbation. Note that the modified Maxwell Eqs.
(1)-(4) can be derived from the Lagrangian density
$\mathcal{L}=\mathcal{L}_{0}+\mathcal{L}_{\Theta}$, where $\mathcal{L}_{0}$ is
the usual Maxwell Lagrangian density and $\mathcal{L}_{\Theta}$ is given by

$\mathcal{L}_{\Theta}=\frac{\alpha}{4\pi^{2}}\frac{\Theta\left(\mathbf{r};\omega\right)}{\mu_{0}c}\mathbf{E}\left(\mathbf{r};\omega\right)\cdot\mathbf{B}\left(\mathbf{r};\omega\right)\;.$
(7)

From a condensed matter perspective, $\Theta$ is referred as the (scalar)
magnetoelectric polarizability Essin , but in field theory, this quantity is
called the axion field, which is the heart of axion electrodynamics Wilczek .
According to the Maxwell Eqs. (1)-(4) and the modified constitutive relations
(5) and (6), this additional $\Theta$ term induces field-dependent effective
charge and current densities which model the topological magnetoelectric
effect present in three-dimensional TIsHasan ; Qi PRB . Let us briefly make
contact with the broader perspective of bi-isotropic materials, which couple
the electric and magnetic fields to each other by means of magnetoelectric
parameters $\chi$ and $\kappa$ characterizing nonreciprocity and chirality
respectively. In this picture, the constitutive relations (5) and (6) can be
regarded as those for a bi-isotropic material
$\mathbf{D}=\varepsilon_{0}\varepsilon\mathbf{E}+(\chi-\mathrm{i}\kappa)\sqrt{\varepsilon_{0}\mu_{0}}\,\mathbf{H}$
and
$\mathbf{B}=\mu_{0}\mu\mathbf{H}+(\chi+\mathrm{i}\kappa)\sqrt{\varepsilon_{0}\mu_{0}}\,\mathbf{E}$
yielding the identifications
$\chi=\alpha\mu\Theta\left(\mathbf{r};\omega\right)/\pi$ and $\kappa=0$.
Therefore, chirality is not present, which is the reason that TIs are
classified as Tellegen media Jiang . Furthermore, the axion coupling adds a
nonreciprocal feature to the TIs which breaks time-reversal symmetry as
mentioned in the Introduction. In the framework of Barron Barron 1 ; Barron 2
, the constitutive relationships for TIs are an instance of false chirality:
the handedness of the medium may be inverted by either space inversion or
time-reversal given that $\chi$ is a time-reversal-odd pseudoscalar Qi PRB .

Combining Eqs. (1)-(4) together with the constitutive relations (5) and (6),
one finds that the frequency components of the electric field obey the
inhomogeneous Helmholtz equation

$\displaystyle\nabla\times\frac{1}{\mu(\mathbf{r};\omega)}\nabla\times\mathbf{E}(\mathbf{r};\omega)-\frac{\omega^{2}}{c^{2}}\varepsilon(\mathbf{r};\omega)\mathbf{E}(\mathbf{r};\omega)$
(8) $\displaystyle-\mathrm{i}\frac{\omega\alpha}{\pi
c}\left[\nabla\Theta(\mathbf{r};\omega)\right]\times\mathbf{E}(\mathbf{r};\omega)=\mathrm{i}\omega\mu_{0}\left[\mathbf{j}(\mathbf{r};\omega)+\mathbf{j}_{N}(\mathbf{r};\omega)\right]\;,$

where
$\mathbf{j}_{N}(\mathbf{r};\omega)=-\mathrm{i}\omega\mathbf{P}_{N}(\mathbf{r};\omega)+\nabla\times\mathbf{M}_{N}(\mathbf{r};\omega)$
is the source term for electromagnetic waves generated by noise fluctuations
within the material. We observe that the last term on the left-hand side
vanishes if the axion coupling is homogeneous,
$\Theta(\mathbf{r};\omega)=\Theta(\omega)$, recovering the propagation of the
electric field in a conventional magnetoelectric. As an immediate consequence,
the propagation of electromagnetic waves will retain their usual properties
within a homogeneous three-dimensional TI. Consequently, the effects due to
the axion coupling will only be felt whenever $\Theta$ presents a spatial
dependency, as is the case at the interface of two materials having different
constant values of $\Theta$, for example. The so-called
$\Theta$-Electrodynamics OJF-LFU-ORT ; AMR-LFU-MC 1 ; AMR-LFU-MC 2  considers
$\Theta$ as a nondynamical field having he necessary topological features to
model the electromagnetic behavior of three-dimensional TIs Qi PRB  as well as
naturally existing magnetoelectric media Hehl ; Landau-Lifshitz  such as Cr2O3
with $\Theta\simeq\pi/36$ Wu ; Hehl-Obukov-Rivera .

With the help of the Green’s function
$\mathbb{G}(\mathbf{r},\mathbf{r}^{\prime};\omega)$ associated with the Eq.
(8), which dictates the dynamics for the electric field, we are able to
compute $\mathbf{E}(\mathbf{r};\omega)$ at any point from an arbitrary
distribution of current sources. The formal solution to the differential
equation (8) reads

$\mathbf{E}(\mathbf{r};\omega)=\mathrm{i}\omega\mu_{0}\int
d^{3}\mathbf{r}\,\mathbb{G}(\mathbf{r},\mathbf{r}^{\prime};\omega)\cdot\left[\mathbf{j}(\mathbf{r};\omega)+\mathbf{j}_{N}(\mathbf{r};\omega)\right]\;.$
(9)

This Green’s function carries all the information of the desired
configuration, which for our case is made up of two semi-infinite
magnetoelectric media separated by a planar interface located at $x=0$,
filling the regions $\mathcal{U}_{1}$ and $\mathcal{U}_{2}$ of the space, as
shown in Fig. 1. At first sight, the choice of such interface at $x=0$ as
opposed to $z=0$ seems to be uncommon but it provides analytical advantages as
Refs. Khan ; Bolotovskii  show. On this work, the axion coupling $\Theta$ will
be taken as a piecewise constant taking the values $\Theta=\Theta_{1}$ in the
region $\mathcal{U}_{1}$ and $\Theta=\Theta_{2}$ in the region
$\mathcal{U}_{2}$. This means that

$\Theta(x)=\Theta_{1}H(x)+\Theta_{2}H(-x)\;,$ (10)

where $H(x)$ is the Heaviside function.

Figure 1:  Two semi-infinite magnetoelectric media with different
magnetoelectric polarizabilities $\Theta_{1}$ and $\Theta_{2}$, having
different permittivities $\varepsilon_{1}$ and $\varepsilon_{2}$, and
different permeabilities $\mu_{1}$ and $\mu_{2}$, and separated by the planar
interface.

The required Green’s function
$\mathbb{G}(\mathbf{r},\mathbf{r}^{\prime};\omega)$ for this configuration can
be obtained from that deduced in Ref. Crosse-Fuchs-Buhmann  by swapping
$x\leftrightarrow z$ in all the indexes and dependences. An important feature
of $\mathbb{G}(\mathbf{r},\mathbf{r}^{\prime};\omega)$ that needs to be
mentioned is its reflective and transmissive behaviors. From Fig. 1 we
recognize two contributions for a field point at $x>0$ and if the source is in
region $\mathcal{U}_{1}$ ($x^{\prime}>0$): one from direct propagation from
the source to the field point, which is described by the free space Green’s
function $\mathbb{G}^{(0)}(\mathbf{r},\mathbf{r}^{\prime};\omega)$, and an
additional one from reflections from the surface, which is dictated by the
reflective part of the Green’s function
$\mathbb{G}^{(1)}(\mathbf{r},\mathbf{r}^{\prime};\omega)$. For $x<0$, the
transmission at the surface is the only contribution that appears and is
described by the transmissive part of the Green’s function
$\mathbb{G}^{(1)}(\mathbf{r},\mathbf{r}^{\prime};\omega)$. This enables us to
split the split the Green’s function into two main parts

$\mathbb{G}(\mathbf{r},\mathbf{r}^{\prime};\omega)=\left\\{\begin{array}[]{cc}\mathbb{G}^{(0)}(\mathbf{r},\mathbf{r}^{\prime};\omega)+\mathbb{G}^{(1)}(\mathbf{r},\mathbf{r}^{\prime};\omega)\;,&x>0\;,\\\
\mathbb{G}^{(1)}(\mathbf{r},\mathbf{r}^{\prime};\omega)\;,&x<0\;,\end{array}\right.$
(11)

each of which can be obtained separately as done in Ref. Crosse-Fuchs-Buhmann
. A second feature of this Green’s function that deserves attention is how it
incorporates the nature of three-dimensional TIs or magnetoelectric media. As
exposed in the Introduction, these materials break time-reversal-symmetry
allowing their classification as noncreciprocal media Fuchs-Crosse-Buhmann .
Moreover, a nonreciprocal medium violates time-reversal symmetry and, hence,
the Lorentz reciprocity principle for the Green’s tensor Buhmann 1 ,

$\mathbb{G}(\mathbf{r},\mathbf{r}^{\prime};\omega)\neq\mathbb{G}^{T}(\mathbf{r}^{\prime},\mathbf{r};\omega)\;.$
(12)

As an immediate consequence of the latter inequality, the symmetry relations
Buhmann 1

$r_{\sigma}^{12}=-r_{\sigma}^{21}\;\;,\;\;\frac{\mu_{2}(\omega)}{k_{x,2}}t_{\sigma}^{21}=\frac{\mu_{1}(\omega)}{k_{x,1}}t_{\sigma}^{12}\;,(\sigma=\mathrm{TM},\mathrm{TE})\;,$
(13)

for the standard Fresnel coefficients are no longer valid for our
configuration. In Eqs. (13), $r_{\sigma}^{12}$ denotes the Fresnel reflection
coefficient and $t_{\sigma}^{12}$ is the transmission one with polarization
$\sigma$, whose superscripts label the medium 1 as an upper layer and the
medium 2 as the lower one or vice versa. Also,
$k_{x,j}=\sqrt{k^{2}_{j}-\mathbf{k}_{\parallel}^{2}}$ stands for the
$x$-component of the wave vector $\mathbf{k}_{j\pm}=(\pm k_{x,j},k_{y},k_{z})$
with wavenumber
$k_{j}=\sqrt{\varepsilon_{j}(\omega)\mu_{j}(\omega)}\,\omega/c$ for $j=1,2$.
From now on, every subscript on a physical quantity will denote that it is
related to the medium 1 or medium 2.

The boundary conditions for the electric and magnetic fields upon assuming
finite time derivatives of the fields in the vicinity of the interface at
$x=0$ as well as the noise terms, can be obtained from the modified Maxwell
equations (1)-(4) and the constitutive relations (5) and (6) lead to the next
boundary conditions:

$\displaystyle\left[\varepsilon_{0}\varepsilon\mathbf{E}_{\perp}\right]_{x=0^{-}}^{x=0^{+}}=\frac{\Delta_{\Theta}\mathbf{B}_{\perp}|_{x=0}}{c\mu_{0}\mu_{1}\mu_{2}}\;,\;\left[\mathbf{B}_{\perp}\right]_{x=0^{-}}^{x=0^{+}}=\mathbf{0}\;,$
(14)
$\displaystyle\left[\frac{\mathbf{B}_{\parallel}}{\mu_{0}\mu}\right]_{x=0^{-}}^{x=0^{+}}=-\frac{\Delta_{\Theta}\mathbf{E}_{\parallel}|_{x=0}}{c\mu_{0}\mu_{1}\mu_{2}}\;,\;\left[\mathbf{E}_{\parallel}\right]_{x=0^{-}}^{x=0^{+}}=\mathbf{0}\;,$
(15)

for vanishing external sources at the interface, where

$\Delta_{\Theta}=\alpha\mu_{1}\mu_{2}(\Theta_{2}-\Theta_{1})/\pi\;.$ (16)

The notation is
$\left[\mathbf{V}\right]_{x=0^{-}}^{x=0^{+}}=\mathbf{V}(x=0^{+})-\mathbf{V}(x=0^{-})$,
$\mathbf{V}\big{|}_{x=0}=\mathbf{V}(x=0)$, where $x=0^{\pm}$ indicates the
limits $x=0\pm\epsilon$, with $b$ a positive real number such that
$\;\epsilon\rightarrow 0$, and $\mathbf{V}$ an arbitrary vector field,
respectively.

When an interface with a regular insulator ($\Theta_{1}=0$) in region
$\mathcal{U}_{1}$ of Fig. 1 and a strong three-dimensional TI located in
region $\mathcal{U}_{2}$ ($\Theta_{2}=\pi$), we have

$\Delta_{\Theta}=\alpha(2m+1),$ (17)

where $m$ is an integer depending on the details of the time-reversal-symmetry
breaking at the interface and $\Delta_{\Theta}$ will be called as topological
parameter.

##  III Electromagnetic fields

Without loss of generality and due to the azimuthal symmetry of our problem,
let us consider a particle with charge $q$ and constant velocity
$v\,\mathbf{e}_{z}$, parallel to the interface defined by the $yz$ plane
($x=0$) where $\mathbf{e}_{z}=(0,0,1)$ as shown in Fig. 1. The external charge
and current densities in frequency space are

$\displaystyle\varrho(\mathbf{r}^{\prime};\omega)$ $\displaystyle=$
$\displaystyle\frac{q}{v}\delta(x^{\prime}-x_{0})\delta(y^{\prime})e^{i\omega
z^{\prime}/v}\;,$ (18) $\displaystyle\mathbf{j}(\mathbf{r}^{\prime};\omega)$
$\displaystyle=$ $\displaystyle
q\delta(x^{\prime}-x_{0})\delta(y^{\prime})e^{i\omega
z^{\prime}/v}\mathbf{e}_{z}\;,$ (19)

where we henceforth assume $x_{0}>0$. Instead of an infinite path for the
charge, i.e., its movement will occur in the interval
$z^{\prime}\in(-\zeta,\zeta)$, with $\zeta\gg v/\omega$ and $\zeta$ is the
half of the total length. This means that the particle travels from left to
right parallel to the interface and never crosses it, as depicted in Fig. 1,
ruling out the presence of transition radiation studied in Ref. OJF-SYB . To
ensure the emission of VC radiation, at least the particle velocity needs to
be greater than the speed of light in one of the two media.

As the current density (19) has a single component, the components of the
Green’s function that matter for this configuration are
$\mathbb{G}^{iz}(\mathbf{r},\mathbf{r}^{\prime};\omega)$ with $i=x,y,z$, whose
explicit forms are given in Appendix A. Using these components, we perform the
indicated convolution of Eq. (9) by substituting the current density (19). The
involved integrals can be simplified by converting to polar coordinates, after
which the angular integrals are computed analytically through integral
representations of the Bessel functions. The resulting Hankel transforms of
the electric field components in the reflective region, upon carrying out the
integral over the parallel wave vector magnitude
$k_{\parallel}=\sqrt{k_{y}^{2}+k_{z}^{2}}$ and the coordinate $z^{\prime}$,
are

$\displaystyle E^{\,x}_{1}(\mathbf{r};\omega)$ $\displaystyle=$
$\displaystyle\mathrm{i}q\omega\mu_{0}\mu_{1}(\omega)$ (20)
$\displaystyle\times\int_{-\zeta}^{\zeta}dz^{\prime}\frac{e^{\mathrm{i}k_{1}R}}{4\pi
R}\left[-\frac{(x-x_{0})(z-z^{\prime})}{R^{2}}\right]e^{\mathrm{i}\frac{\omega
z^{\prime}}{v}}$
$\displaystyle+\frac{\mathrm{i}q\omega\mu_{0}\mu_{1}(\omega)}{4\pi
k_{1}^{2}}\int_{-\zeta}^{\zeta}dz^{\prime}e^{\mathrm{i}\frac{\omega
z^{\prime}}{v}}\frac{(z-z^{\prime})}{R_{\parallel}}\frac{\partial\mathcal{I}_{1}}{\partial
R_{\parallel}}$
$\displaystyle+\frac{\mathrm{i}q\omega\mu_{0}\mu_{1}(\omega)}{4\pi
k_{1}}\int_{-\zeta}^{\zeta}dz^{\prime}e^{\mathrm{i}\frac{\omega
z^{\prime}}{v}}\frac{y}{R_{\parallel}}\frac{\partial\mathcal{I}_{2}}{\partial
R_{\parallel}}\;,$ $\displaystyle E^{\,y}_{1}(\mathbf{r};\omega)$
$\displaystyle=$
$\displaystyle-\mathrm{i}q\omega\mu_{0}\mu_{1}(\omega)\int_{-\zeta}^{\zeta}dz^{\prime}\frac{e^{\mathrm{i}k_{1}R}}{4\pi
R}\frac{y(z-z^{\prime})}{R^{2}}e^{\mathrm{i}\frac{\omega z^{\prime}}{v}}$ (21)
$\displaystyle+\frac{q\omega\mu_{0}\mu_{1}(\omega)}{4\pi
k_{1}}\int_{-\zeta}^{\zeta}dz^{\prime}e^{\mathrm{i}\frac{\omega
z^{\prime}}{v}}\mathcal{I}_{3}\;,$ $\displaystyle
E^{\,z}_{1}(\mathbf{r};\omega)$ $\displaystyle=$
$\displaystyle-\mathrm{i}q\omega\mu_{0}\mu_{1}(\omega)$ (22)
$\displaystyle\times\int_{-\zeta}^{\zeta}dz^{\prime}\frac{e^{\mathrm{i}k_{1}R}}{4\pi
R}\left[1-\frac{(z-z^{\prime})^{2}}{R^{2}}\right]e^{\mathrm{i}\frac{\omega
x^{\prime}}{v}}$
$\displaystyle+\frac{q\omega\mu_{0}\mu_{1}(\omega)}{4\pi}\int_{-\zeta}^{\zeta}dz^{\prime}\frac{e^{\mathrm{i}\frac{\omega
z^{\prime}}{v}}}{R_{\parallel}}\frac{\partial\mathcal{I}_{4}}{\partial
R_{\parallel}}$ $\displaystyle+\frac{q\omega\mu_{0}\mu_{1}(\omega)}{4\pi
k_{1}^{2}}\int_{-\zeta}^{\zeta}dz^{\prime}\frac{e^{\mathrm{i}\frac{\omega
z^{\prime}}{v}}}{R_{\parallel}}\frac{\partial\mathcal{I}_{5}}{\partial
R_{\parallel}}$ $\displaystyle+\frac{q\omega\mu_{0}\mu_{1}(\omega)}{4\pi
k_{1}^{2}}\int_{-\zeta}^{\zeta}dz^{\prime}e^{\mathrm{i}\frac{\omega
z^{\prime}}{v}}\mathcal{I}_{6}\;,$

where $R_{\parallel}=\sqrt{y^{2}+(z-z^{\prime})^{2}}$,
$R^{2}=(x-x_{0})^{2}+R_{\parallel}^{2}$ and the following integrals were
defined

$\displaystyle\mathcal{I}_{1}$
$\displaystyle=\int_{0}^{\infty}dk_{\parallel}k_{\parallel}R_{\mathrm{TM,TM}}^{12}(k_{\parallel})J_{0}(k_{\parallel}R_{\parallel})e^{\mathrm{i}k_{x,1}(x+x_{0})}\;,$
(23)
$\mathcal{I}_{2}=\int_{0}^{\infty}\frac{dk_{\parallel}k_{\parallel}}{k_{x,1}}R_{\mathrm{TM,TE}}^{12}(k_{\parallel})J_{0}(k_{\parallel}R_{\parallel})e^{\mathrm{i}k_{x,1}(x+x_{0})}\;,$
(24)
$\mathcal{I}_{3}=\int_{0}^{\infty}dk_{\parallel}k_{\parallel}R_{\mathrm{TE,TM}}^{12}(k_{\parallel})J_{0}(k_{\parallel}R_{\parallel})e^{\mathrm{i}k_{x,1}(x+x_{0})}\;,$
(25)
$\mathcal{I}_{4}=\int_{0}^{\infty}\frac{dk_{\parallel}}{k_{\parallel}k_{x,1}}R_{\mathrm{TE,TE}}^{12}(k_{\parallel})J_{0}(k_{\parallel}R_{\parallel})e^{\mathrm{i}k_{x,1}(x+x_{0})}\;,$
(26)
$\mathcal{I}_{5}=\int_{0}^{\infty}\frac{dk_{\parallel}k_{x,1}}{k_{\parallel}}R_{\mathrm{TM,TM}}^{12}(k_{\parallel})J_{0}(k_{\parallel}R_{\parallel})e^{\mathrm{i}k_{x,1}(x+x_{0})}\;,$
(27)
$\mathcal{I}_{6}=\int_{0}^{\infty}dk_{\parallel}k_{\parallel}k_{x,1}R_{\mathrm{TM,TM}}^{12}(k_{\parallel})J_{0}(k_{\parallel}R_{\parallel})e^{\mathrm{i}k_{x,1}(x+x_{0})}\;.$
(28)

Analogously for the transmissive region, we have

$\displaystyle E^{\,x}_{2}(\mathbf{r};\omega)$ $\displaystyle=$
$\displaystyle\frac{\mathrm{i}q\omega\mu_{0}\mu_{1}(\omega)}{4\pi
k_{1}k_{2}}\int_{-\zeta}^{\zeta}dz^{\prime}e^{\mathrm{i}\frac{\omega
z^{\prime}}{v}}\frac{(z-z^{\prime})}{R_{\parallel}}\frac{\partial\mathcal{I}_{7}}{\partial
R_{\parallel}}$ (29)
$\displaystyle+\frac{\mathrm{i}q\omega\mu_{0}\mu_{1}(\omega)}{4\pi
k_{2}}\int_{-\zeta}^{\zeta}dz^{\prime}e^{\mathrm{i}\frac{\omega
z^{\prime}}{v}}\frac{y}{R_{\parallel}}\frac{\partial\mathcal{I}_{8}}{\partial
R_{\parallel}}\;,$ $\displaystyle E^{\,y}_{2}(\mathbf{r};\omega)$
$\displaystyle=$ $\displaystyle\frac{q\omega\mu_{0}\mu_{1}(\omega)}{4\pi
k_{1}}\int_{-\zeta}^{\zeta}dz^{\prime}\frac{e^{\mathrm{i}\frac{\omega
z^{\prime}}{v}}}{R_{\parallel}}\frac{\partial\mathcal{I}_{9}}{\partial
R_{\parallel}}$ (30) $\displaystyle+\frac{q\omega\mu_{0}\mu_{1}(\omega)}{4\pi
k_{1}}\int_{-\zeta}^{\zeta}dz^{\prime}e^{\mathrm{i}\frac{\omega
z^{\prime}}{v}}\mathcal{I}_{10}$
$\displaystyle+\frac{q\omega\mu_{0}\mu_{1}(\omega)}{4\pi
k_{2}}\int_{-\zeta}^{\zeta}dz^{\prime}\frac{e^{\mathrm{i}\frac{\omega
z^{\prime}}{v}}}{R_{\parallel}}\frac{\partial\mathcal{I}_{11}}{\partial
R_{\parallel}}\;,$ $\displaystyle E^{\,z}_{2}(\mathbf{r};\omega)$
$\displaystyle=$
$\displaystyle\frac{q\omega\mu_{0}\mu_{1}(\omega)}{4\pi}\int_{-\zeta}^{\zeta}dz^{\prime}\frac{e^{\mathrm{i}\frac{\omega
z^{\prime}}{v}}}{R_{\parallel}}\frac{\partial\mathcal{I}_{12}}{\partial
R_{\parallel}}$ (31) $\displaystyle-\frac{q\omega\mu_{0}\mu_{1}(\omega)}{4\pi
k_{1}k_{2}}\int_{-\zeta}^{\zeta}dz^{\prime}\frac{e^{\mathrm{i}\frac{\omega
z^{\prime}}{v}}}{R_{\parallel}}\frac{\partial\mathcal{I}_{13}}{\partial
R_{\parallel}}$ $\displaystyle-\frac{q\omega\mu_{0}\mu_{1}(\omega)}{4\pi
k_{1}k_{2}}\int_{-\zeta}^{\zeta}dz^{\prime}e^{\mathrm{i}\frac{\omega
z^{\prime}}{v}}\mathcal{I}_{14}\;,$

where the following integrals were also defined

$\mathcal{I}_{7}=\int_{0}^{\infty}dk_{\parallel}k_{\parallel}T_{\mathrm{TM,TM}}^{12}(k_{\parallel})J_{0}(k_{\parallel}R_{\parallel})e^{\mathrm{i}k_{x,2}|x|+\mathrm{i}k_{x,1}x_{0}}\;,$
(32)
$\mathcal{I}_{8}=\int_{0}^{\infty}\frac{dk_{\parallel}k_{\parallel}}{k_{x,1}}T_{\mathrm{TM,TE}}^{12}(k_{\parallel})J_{0}(k_{\parallel}R_{\parallel})e^{\mathrm{i}k_{x,2}|x|+\mathrm{i}k_{x,1}x_{0}}\;,$
(33)
$\mathcal{I}_{9}=\int_{0}^{\infty}\frac{dk_{\parallel}}{k_{\parallel}}T_{\mathrm{TE,TM}}^{12}(k_{\parallel})J_{0}(k_{\parallel}R_{\parallel})e^{\mathrm{i}k_{x,2}|x|+\mathrm{i}k_{x,1}x_{0}}\;,$
(34)
$\mathcal{I}_{10}=\int_{0}^{\infty}dk_{\parallel}k_{\parallel}T_{\mathrm{TE,TM}}^{12}(k_{\parallel})J_{0}(k_{\parallel}R_{\parallel})e^{\mathrm{i}k_{x,2}|x|+\mathrm{i}k_{x,1}x_{0}}\;,$
(35)
$\mathcal{I}_{11}=\int_{0}^{\infty}\frac{dk_{\parallel}k_{x,2}}{k_{\parallel}k_{x,1}}T_{\mathrm{TM,TE}}^{12}(k_{\parallel})J_{0}(k_{\parallel}R_{\parallel})e^{\mathrm{i}k_{x,2}|x|+\mathrm{i}k_{x,1}x_{0}}\;,$
(36)
$\mathcal{I}_{12}=\int_{0}^{\infty}\frac{dk_{\parallel}}{k_{\parallel}k_{x,1}}T_{\mathrm{TE,TE}}^{12}(k_{\parallel})J_{0}(k_{\parallel}R_{\parallel})e^{\mathrm{i}k_{x,2}|x|+\mathrm{i}k_{x,1}x_{0}}\;,$
(37)
$\mathcal{I}_{13}=\int_{0}^{\infty}\frac{dk_{\parallel}k_{x,2}}{k_{\parallel}}T_{\mathrm{TM,TM}}^{12}(k_{\parallel})J_{0}(k_{\parallel}R_{\parallel})e^{\mathrm{i}k_{x,2}|x|+\mathrm{i}k_{x,1}x_{0}}\;,$
(38)
$\mathcal{I}_{14}=\int_{0}^{\infty}dk_{\parallel}k_{\parallel}k_{x,2}T_{\mathrm{TM,TM}}^{12}(k_{\parallel})J_{0}(k_{\parallel}R_{\parallel})e^{\mathrm{i}k_{x,2}|x|+\mathrm{i}k_{x,1}x_{0}}\;.$
(39)

In the components (20)-(22) and (29)-(31), terms of order higher than $R^{-2}$
were neglected and the eight modified Fresnel coefficients required for the
electric field components are given as follows Crosse-Fuchs-Buhmann :

$\displaystyle R_{\mathrm{TE,TE}}^{12}$ $\displaystyle=$
$\displaystyle\frac{(\mu_{2}k_{x,1}-\mu_{1}k_{x,2})\Omega_{\varepsilon}-\Delta_{\Theta}^{2}k_{x,1}k_{x,2}}{(\varepsilon_{2}k_{x,1}+\varepsilon_{1}k_{x,2})\Omega_{\mu}+\Delta_{\Theta}^{2}k_{x,1}k_{x,2}}\;,$
(40) $\displaystyle R_{\mathrm{TM,TE}}^{12}$ $\displaystyle=$
$\displaystyle\frac{-2\mu_{2}n_{1}k_{x,1}k_{x,2}\Delta_{\Theta}}{(\varepsilon_{2}k_{x,1}+\varepsilon_{1}k_{x,2})\Omega_{\mu}+\Delta_{\Theta}^{2}k_{x,1}k_{x,2}}\;,$
(41) $\displaystyle T_{\mathrm{TE,TE}}^{12}$ $\displaystyle=$
$\displaystyle\frac{2\mu_{2}k_{x,1}\Omega_{\varepsilon}}{(\varepsilon_{2}k_{x,1}+\varepsilon_{1}k_{x,2})\Omega_{\mu}+\Delta_{\Theta}^{2}k_{x,1}k_{x,2}},\;\;$
(42) $\displaystyle T_{\mathrm{TM,TE}}^{12}$ $\displaystyle=$
$\displaystyle\frac{2\mu_{2}n_{2}k_{x,1}^{2}\Delta_{\Theta}}{(\varepsilon_{2}k_{x,1}+\varepsilon_{1}k_{x,2})\Omega_{\mu}+\Delta_{\Theta}^{2}k_{x,1}k_{x,2}}\;,$
(43) $\displaystyle R_{\mathrm{TM,TM}}^{12}$ $\displaystyle=$
$\displaystyle\frac{(\varepsilon_{2}k_{x,1}-\varepsilon_{1}k_{x,2})\Omega_{\mu}+\Delta_{\Theta}^{2}k_{x,1}k_{x,2}}{(\varepsilon_{2}k_{x,1}+\varepsilon_{1}k_{x,2})\Omega_{\mu}+\Delta_{\Theta}^{2}k_{x,1}k_{x,2}}\;,$
(44) $\displaystyle R_{\mathrm{TE,TM}}^{12}$ $\displaystyle=$ $\displaystyle
R_{\mathrm{TM,TE}}^{12}\;,$ (45) $\displaystyle T_{\mathrm{TM,TM}}^{12}$
$\displaystyle=$
$\displaystyle\frac{n_{2}}{n_{1}}\frac{2\varepsilon_{1}k_{x,1}\Omega_{\mu}}{(\varepsilon_{2}k_{x,1}+\varepsilon_{1}k_{x,2})\Omega_{\mu}+\Delta_{\Theta}^{2}k_{x,1}k_{x,2}},\;\;\;\;$
(46) $\displaystyle T_{\mathrm{TE,TM}}^{12}$ $\displaystyle=$
$\displaystyle-\frac{n_{1}k_{x,2}}{n_{2}k_{x,1}}T_{\mathrm{TM,TE}}^{12}\;,$
(47)

where
$\Omega_{\varepsilon}=\mu_{1}\mu_{2}(k_{x,1}\varepsilon_{2}+k_{x,2}\varepsilon_{1})$,
$\Omega_{\mu}=\mu_{1}\mu_{2}(k_{x,1}\mu_{2}+k_{x,2}\mu_{1})$, and recalling
that $\Delta_{\Theta}$ was defined in Eq. (16) and specified for TIs in Eq.
(17). For these modified Fresnel coefficients the notation for the
superscripts described for the usual Fresnel coefficients (13) is employed,
while the subscripts show explicitly a mixture between polarizations TE and TM
Crosse-Fuchs-Buhmann . Of course, the coefficients (40)-(47) reduce properly
to the standard ones when $\Delta_{\Theta}$ is turned off.

Our next task consists of solving the integrals over $k_{\parallel}$. Focusing
on the radiation emitted by the particle, we will implement the far-field
approximation retaining for the moment that $x_{0}$ could be of arbitrary
size. To this aim we employ a steepest descent method Chew ; Mandel ; Chew art
; Wait . By means of this method, whose application with full details to the
involved integrals is in Appendix B, we obtain the following results after
discarding terms of order higher than $R^{-2}$ for the reflected electric
field

$\displaystyle\mathbf{E}_{1}(\mathbf{r};\omega)=\frac{\mathrm{i}q\omega\mu_{0}\mu_{1}(\omega)\mathcal{K}_{1}(z/R_{0},\omega)}{4\pi
R_{0}}e^{\mathrm{i}k_{1}R_{0}}$
$\displaystyle\times\left(\begin{array}[]{c}-(x-x_{0})z/R^{2}_{0}\\\
-yz/R^{2}_{0}\\\ 1-z^{2}/R^{2}_{0}\end{array}\right)$ (51)
$\displaystyle+\frac{\mathrm{i}q\omega\mu_{0}\mu_{1}(\omega)\tilde{\mathcal{K}}_{1}(z/R_{1},\omega)}{4\pi
R_{1}}e^{\mathrm{i}k_{1}R_{1}}$
$\displaystyle\times\left[R_{\mathrm{TM,TM}}^{12}(k_{1}\rho/R_{1},\Delta_{\Theta})\left(\begin{array}[]{c}(x+x_{0})z/R_{1}^{2}\\\
0\\\ -(x+x_{0})^{2}/R_{1}^{2}\end{array}\right)\right.$ (55)
$\displaystyle\left.+R_{\mathrm{TE,TM}}^{12}(k_{1}\rho/R_{1},\Delta_{\Theta})\left(\begin{array}[]{c}y/R_{1}\\\
-(x+x_{0})/R_{1}\\\ 0\end{array}\right)\right]$ (59)
$\displaystyle+\frac{q\omega\mu_{0}\mu_{1}(\omega)}{4\pi
k_{1}}\sqrt{\frac{2\pi\mathrm{i}\varkappa}{\rho}}e^{\mathrm{i}\varkappa\rho+\mathrm{i}\sqrt{k_{1}^{2}-\varkappa^{2}}(x+x_{0})}\mathcal{M}(z/\rho,\omega)$
$\displaystyle\times\left[\frac{\mathrm{Res}\left(R_{\mathrm{TM,TM}}^{12};\varkappa\right)}{k_{1}}\left(\begin{array}[]{c}-\varkappa
z/\rho\\\ 0\\\ \sqrt{k_{1}^{2}-\varkappa^{2}}\end{array}\right)\right.$ (63)
$\displaystyle+\left.\mathrm{Res}\left(R_{\mathrm{TE,TM}}^{12};\varkappa\right)\left(\begin{array}[]{c}-\varkappa
y/(\rho\sqrt{k_{1}^{2}-\varkappa^{2}})\\\ 1\\\ 0\end{array}\right)\right]\;,$
(67)

where now $R_{0}=\sqrt{(x-x_{0})^{2}+y^{2}+z^{2}}$,
$R_{1}=\sqrt{(x+x_{0})^{2}+y^{2}+z^{2}}$, $\rho=\sqrt{y^{2}+z^{2}}$ and

$\displaystyle\mathcal{K}_{1}$ $\displaystyle=$
$\displaystyle\int_{-\zeta}^{\zeta}dz^{\prime}e^{\mathrm{i}\frac{\omega
z^{\prime}}{v}\left(1-\frac{vn_{1}z}{cR_{0}}\right)}=\frac{2\sin\left(\zeta\Xi_{1}\right)}{\Xi_{1}}\;,$
(68) $\displaystyle\tilde{\mathcal{K}}_{1}$ $\displaystyle=$
$\displaystyle\int_{-\zeta}^{\zeta}dz^{\prime}e^{\mathrm{i}\frac{\omega
z^{\prime}}{v}\left(1-\frac{vn_{1}z}{cR_{1}}\right)}=\frac{2\sin\left(\zeta\tilde{\Xi}_{1}\right)}{\tilde{\Xi}_{1}}\;,$
(69) $\displaystyle\mathcal{M}$ $\displaystyle=$
$\displaystyle\int_{-\zeta}^{\zeta}dz^{\prime}\exp\left[\mathrm{i}z^{\prime}\left(\frac{\omega}{v}-\varkappa\frac{z}{\rho}\right)\right]\;,$
(70)

are the resulting integrals over $z^{\prime}$ with

$\Xi_{1}=\frac{\omega}{v}\left(1-\frac{vn_{1}z}{cR_{0}}\right)\;,\;\tilde{\Xi}_{1}=\frac{\omega}{v}\left(1-\frac{vn_{1}z}{cR_{1}}\right)\;.$
(71)

In Eq. (67), $\mathrm{Res}$ is the residue of the indicated Fresnel
coefficient at the simple pole $\varkappa$, which is the only physical root of
the polynomial of degree 4 in $k_{\parallel}$ arising from the common
denominator for all modified Fresnel coefficients and has units of wave
number.

For the transmitted field, with $x_{0}\ll x$, we find

$\displaystyle\mathbf{E}_{2}(\mathbf{r};\omega)=\frac{\mathrm{i}q\omega\mu_{0}\mu_{1}(\omega)k_{2}\mathcal{K}_{2}(z/r,\omega)}{4\pi
k_{1}r}e^{\mathrm{i}k_{2}r}e^{\mathrm{i}\xi x_{0}}$
$\displaystyle\times\left\\{T_{\mathrm{TM,TM}}^{12}(k_{2}\rho/r,\Delta_{\Theta})\left(\begin{array}[]{c}-xz/r^{2}\\\
0\\\ x^{2}/r^{2}\end{array}\right)\right.$ (75)
$\displaystyle\left.+T_{\mathrm{TE,TM}}^{12}(k_{2}\rho/r,\Delta_{\Theta})\left(\begin{array}[]{c}-y/r\\\
x/r\\\ 0\end{array}\right)\right\\}$ (79)
$\displaystyle+\frac{q\omega\mu_{0}\mu_{1}(\omega)}{4\pi
k_{1}}\sqrt{\frac{2\pi\mathrm{i}\varkappa}{\rho}}e^{\mathrm{i}\varkappa\rho+\mathrm{i}\sqrt{k_{2}^{2}-\varkappa^{2}}|x|}e^{\mathrm{i}\sqrt{k_{1}^{2}-\varkappa^{2}}x_{0}}$
$\displaystyle\times\mathcal{M}(z/\rho,\omega)\left[\frac{\mathrm{Res}\left(T_{\mathrm{TM,TM}}^{12};\varkappa\right)}{k_{2}}\left(\begin{array}[]{c}-\varkappa
z/\rho\\\ 0\\\ \sqrt{k_{2}^{2}-\varkappa^{2}}\end{array}\right)\right.$ (83)
$\displaystyle+\left.\mathrm{Res}\left(T_{\mathrm{TE,TM}}^{12};\varkappa\right)\left(\begin{array}[]{c}-\varkappa
y/(\rho\sqrt{k_{2}^{2}-\varkappa^{2}})\\\ 1\\\ 0\end{array}\right)\right],\;$
(87)

where $r=\sqrt{x^{2}+y^{2}+z^{2}}$,
$\xi=\sqrt{k_{1}^{2}-k_{2}^{2}(1-x^{2}/r^{2})}$, $\mathcal{M}$ is given by Eq.
(70) but must be now considered in the lower medium and

$\mathcal{K}_{2}=\int_{-\zeta}^{\zeta}dz^{\prime}e^{\mathrm{i}\frac{\omega
z^{\prime}}{v}\left(1-\frac{vn_{2}z}{cr}\right)}=\frac{2\sin\left(\zeta\Xi_{2}\right)}{\Xi_{2}}\;,$
(88)

is the corresponding integral over $z^{\prime}$ with

$\Xi_{2}=\frac{\omega}{v}\left(1-\frac{vn_{2}z}{cr}\right)\;.$ (89)

At this stage becomes necessary to discuss the physical meaning of the
electric field given by Eqs. (67) and (87). First of all, we appreciate that
two kind of waves contribute to the electric field in both media: spherical
and evanescent surface waves. As usual, the spherical-wave contribution decays
with the inverse of the distance, where we identify two terms: one associated
to the free-space contribution plus one related to the response of the
interface proportional to the modified Fresnel coefficients
$R_{\mathrm{TM,TM}}^{12}$ or $T_{\mathrm{TM,TM}}^{12}$ both being quadratic
contributions of the topological parameter $\Delta_{\Theta}$. This response
has a second contribution proportional to the crossed Fresnel coefficients
$R_{\mathrm{TE,TM}}^{12}$ in medium 1 or $T_{\mathrm{TE,TM}}^{12}$ in medium
2, which leads to purely topological contributions because they are linear in
the topological parameter $\Delta_{\Theta}$ and are absent in the standard
case $\Delta_{\Theta}=0$. An important feature of this field is the VC cone
arising from the integrals over $z^{\prime}$, which lead to the functions
$\mathcal{K}_{1}$ (68), $\tilde{\mathcal{K}}_{1}$ (69) and $\mathcal{K}_{2}$
(88). From their corresponding arguments $\Xi_{1}$ (71), $\tilde{\Xi}_{1}$
(71) and $\Xi_{2}$ (89) of the sinus function, we can define the Čerenkov
angle that determines the VC cone aperture. Namely, we will have the following
three Čerenkov angles:

$\displaystyle\cos\theta_{1}^{C}=\frac{z}{R}=\frac{c}{vn_{1}}$ ,
$\displaystyle\cos\tilde{\theta}_{1}^{C}=\frac{z}{R_{1}}=\frac{c}{vn_{1}}\;,$
(90) $\displaystyle\cos\theta_{2}^{C}$ $\displaystyle=$
$\displaystyle\frac{z}{r}=\frac{c}{vn_{2}}\;,$ (91)

which must lie in the interval $[0,\pi/2)$ otherwise there will not be VC
radiation. Note that the Čerenkov angles of the free contribution and of the
reflected one are equal in magnitude but different functionally because the
reflected one arises from the electric field generated by the image charge.
These angles differ from the transmissive one as found previously in Ref.
Bolotovskii . This fact can be understood in simple terms when we recall the
behavior of light in front of an interface, meaning that $\theta_{1}^{C}$ is
the incidence angle of the VC radiation due to the free term of the electric
field, $\tilde{\theta}_{1}^{C}$ is the reflected one both for the
$R_{\mathrm{TM,TM}}^{12}$ and $R_{\mathrm{TE,TM}}^{12}$ terms, and the
$\theta_{2}^{C}$ is the refracted one related to $T_{\mathrm{TM,TM}}^{12}$ and
$T_{\mathrm{TE,TM}}^{12}$. As a consequence of the Cherenkov angles given by
Eqs. (90) and (91), for this configuration a non-symmetric VC cone can arise
Bolotovskii  and the reflective part or the transmissive part of the VC cone
can be individually suppressed. The phases at both sides of the interface
differ as may be expected, being $e^{\mathrm{i}k_{1}R_{0}}$ and
$e^{\mathrm{i}k_{1}R_{1}}$ due to the reflection response from the interface.
On the other hand, $e^{\mathrm{i}\xi x_{0}}$ accounts for the radiation
transmitted through the interface, which could become a decaying exponential
if $n_{1}<n_{2}$ meaning that the radiation would be dramatically attenuated
in a similar way as happens to the dipolar radiation in front a single
interface Novotny . In contrast to the previously studied case where this
phases difference was also found in the perpendicular motion towards the same
interface but with same permittivities OJF-LFU-ORT , here the possibility of a
reversed VC radiation is ruled out because no phase allows negative angles.
Physically speaking, this can be understood because the charge does not cross
the interface at any point so the VC cone cannot be reflected backwards with
respect to the charge’s trajectory but only reflected forwards in the same
direction of the particle’s trajectory. Finally, we observe that the TE,TE
polarization is no more present because it is of higher order than $r^{-2}$.

Apart from the spherical-wave contribution, we find surface waves in Eqs. (67)
and (87). For the sake of discussion, we present their explicit form for the
case of non magnetic media $\mu_{1}=\mu_{2}=1$. At order
$\Delta_{\Theta}^{2}$, the reflected field is given by

$\displaystyle\mathbf{E}_{1}^{\;\mathrm{surface}}(\mathbf{r};\omega)=\frac{q\omega\mu_{0}}{4\pi
n_{1}}\sqrt{\frac{2\pi\mathrm{i}\omega}{\rho
c}}e^{\mathrm{i}\varkappa\rho+\mathrm{i}\sqrt{k_{1}^{2}-\varkappa^{2}}(x+x_{0})}$
$\displaystyle\times\mathcal{M}(z/\rho,\omega)\left[\frac{\mathcal{F}_{1}(\varepsilon_{1},\varepsilon_{2},\Delta_{\Theta})}{n_{1}}\left(-\frac{z}{\rho}\mathbf{e}_{x}+\mathbf{e}_{z}\right)\right.$
$\displaystyle+\left.\mathcal{F}_{2}(\varepsilon_{1},\varepsilon_{2},\Delta_{\Theta})\left(-\frac{y}{\rho}\mathbf{e}_{x}+\mathbf{e}_{y}\right)\right]\;,$
(92)

with $\mathcal{M}$ taking the next form:

$\displaystyle\mathcal{M}$ $\displaystyle=$
$\displaystyle\int_{-\zeta}^{\zeta}dz^{\prime}\exp\left\\{\mathrm{i}\frac{\omega
z^{\prime}}{v}\left[1-\frac{vn_{1}}{c}\sin\tilde{\theta}_{B}\frac{z}{\rho}\right]\right\\}\;,$
(93) $\displaystyle=$
$\displaystyle\frac{2\sin\left(\zeta\Lambda\right)}{\Lambda}\;,$

where

$\displaystyle\Lambda$ $\displaystyle=$
$\displaystyle\frac{\omega}{v}\left[1-\frac{vn_{1}}{c}\sin\tilde{\theta}_{B}\frac{z}{\rho}\right]\;,$
(94) $\displaystyle\varkappa$ $\displaystyle=$ $\displaystyle
k_{1}\sin\tilde{\theta}_{B}\;,$ (95) $\displaystyle\sin\tilde{\theta}_{B}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{\varepsilon_{2}}{\varepsilon_{1}+\varepsilon_{2}}}\left[1+\frac{\Delta_{\Theta}^{2}\varepsilon_{1}\varepsilon_{2}}{(\varepsilon_{1}-\varepsilon_{2})^{2}(\varepsilon_{1}+\varepsilon_{2})}\right].$
(96)

The transmitted field can be written in the following fashion:

$\displaystyle\mathbf{E}_{2}^{\;\mathrm{surface}}(\mathbf{r};\omega)=\frac{q\omega\mu_{0}}{4\pi
n_{1}}\sqrt{\frac{2\pi\mathrm{i}\omega}{\rho
c}}e^{\mathrm{i}\varkappa\rho+\mathrm{i}\sqrt{k_{2}^{2}-\varkappa^{2}}|x|}$
$\displaystyle\times
e^{\mathrm{i}\sqrt{k_{1}^{2}-\varkappa^{2}}x_{0}}\mathcal{M}(z/\rho,\omega)$
$\displaystyle\times\left[\frac{\mathcal{F}_{1}(\varepsilon_{1},\varepsilon_{2},\Delta_{\Theta})}{n_{2}}\left(-\frac{z}{\rho}\mathbf{e}_{x}+\mathbf{e}_{z}\right)\right.$
$\displaystyle+\left.\mathcal{F}_{2}(\varepsilon_{1},\varepsilon_{2},\Delta_{\Theta})\left(-\frac{y}{\rho}\mathbf{e}_{x}+\mathbf{e}_{y}\right)\right]\;,$
(97)

where $\mathcal{M}$ was just above defined in Eq. (93). For Eqs. (92) and
(97), $\mathcal{F}_{j}$ with $j=1,2$ are lengthy functions depending only on
$\varepsilon_{1},\varepsilon_{2},\Delta_{\Theta}$ and their explicit
expressions can be found in the Appendix C.

Let us now focus on the physics behind the surface-wave contribution. At first
sight, we can split it into two kinds: cylindrical decaying as $\rho^{-1/2}$
and conical $\rho^{-3/2}$ Chew , which differs from the lateral waves with a
$\rho^{-2}$ dependence found for the transition radiation case in Ref. OJF-SYB
. Here, the mathematical origin of these waves comes from the fact that all
the relevant integrals over $k_{\parallel}$ of the electric field exhibit a
simple pole at $\varkappa$. However, the cylindrical waves propagate along the
plane parallel to the interface and can be physically interpreted as surface
plasmon modes. Meanwhile, the conical ones decay rather faster than the
previous ones and are the excited modes oscillating in the $\mathbf{e}_{x}$
direction perpendicular to the interface. Naturally, it is expected that for a
charge traveling far away from the interface both contributions will not be
excited. Furthermore, we see that the contributions modulated by
$\mathcal{F}_{1}$ (175) and $\mathcal{F}_{2}$ (176) are of topological origin
due to their quadratic dependence in the topological parameter
$\Delta_{\Theta}$ meaning that they are absent in the standard case
$\Delta_{\Theta}=0$. Another significant consequence of the topological
parameter $\Delta_{\Theta}$ is the modification of the Brewster angle
$\tilde{\theta}_{B}$ (96), as studied in the work Chang-Yang , which dictates
the propagation of both surface waves by means of $\varkappa$ (95).
Interestingly, in order to exist, we observe that the surface waves must
satisfy the condition

$\frac{vn_{1}}{c}\sin\tilde{\theta}_{B}\frac{z}{\sqrt{y^{2}+z^{2}}}=1,$ (98)

which comes directly from the function $\mathcal{M}$ (93). Equation (98) can
be regarded as a different VC condition where the modified Brewster angle (96)
has a key role. For the case $\varkappa>k_{1,2}$ we found that the conical and
the cylindrical waves can be evanescent because the corresponding squared root
becomes imaginary providing an exponential decay of both surface waves.
Otherwise, if $\varkappa<k_{1,2}$, they will only decay algebraically.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2:  The electric field pattern (real part) in the positive $x-z$ plane
(reflection region) for a charge particle with single frequency
$\omega=6.0450\times 10^{14}\,\,\mathrm{s}^{-1}$, constant velocity
$v=0.95\,c$ parallel to the $x$ axis at a height $x_{0}=0.25\,\mathrm{m}$ from
an interface constituted by standard dielectrics with $\varepsilon_{1}=1.2$,
$\mu_{1}=1$ at the upper layer and another one with $\varepsilon_{2}=4$,
$\mu_{2}=1$ at the lower one as depicted in plots (a), (b) and (c). The plots
(d), (e) and (f) show again the electric field for an interface conformed by
the same upper medium but now with the topological insulator TlBiSe2 with
$\varepsilon_{2}=4$, $\mu_{2}=1$ and $\Delta_{\Theta}=11\alpha$ a the lower
one. The electric field is dimensionless. The charge moves from the bottom to
the top.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3:  The electric field pattern (real part) in the negative $x-z$ plane
(transmission region) for a charge particle with single frequency
$\omega=6.0450\times 10^{14}\,\,\mathrm{s}^{-1}$, constant velocity $v=0.9\,c$
parallel to the $x$ axis at a height $x_{0}=1\times 10^{-4}\,\mathrm{m}$ from
an interface constituted by standard dielectrics with $\varepsilon_{1}=1.2$,
$\mu_{1}=1$ at the upper layer and another one with $\varepsilon_{2}=4$,
$\mu_{2}=1$ at the lower one as depicted in plots (a), (b) and (c). The plots
(d), (e) and (f) show again the electric field for an interface conformed by
the same upper medium but now with the topological insulator TlBiSe2 with
$\varepsilon_{2}=4$, $\mu_{2}=1$ and $\Delta_{\Theta}=11\alpha$ a the lower
one. The electric field is dimensionless. The charge moves from the bottom to
the top.

To conclude this section, and for illustrative purposes, we show the radiation
part of electric field at both sides of the interface two in Figs. 2 and 3.
The first column of Figs. 2 and 3 displays to the $x$ component of the
electric field, the second and the third columns feature the $y$ and $z$
components respectively. The first row Figs. 2 and 3 exhibits the electric
field components for a bi-layer configuration constituted by two standard
dielectrics with $\varepsilon_{1}=1.2$, $\mu_{1}=1$ and $\Delta_{\Theta}=0$ at
the upper layer and another one with $\varepsilon_{2}=4$, $\mu_{2}=1$ and
$\Delta_{\Theta}=0$ at the lower one. The second row of these figures displays
again the electric field components but now for an interface conformed by the
same standard dielectric with $\varepsilon_{1}=1.2$, $\mu_{1}=1$ and
$\Delta_{\Theta}=0$ at the upper layer and the topological insulator TlBiSe2
with $\varepsilon_{2}=4$, $\mu_{2}=1$ and $\Delta_{\Theta}=11\alpha$ at the
lower one TlBiSe2 . The charge moves with a velocity $v=0.95\,c$ from the
bottom to the top at a distance $x_{0}=0.25\,\mathrm{m}$ to the interface
located at $x=0$ in Fig. 2 and in Fig. 3 with a velocity $v=0.9\,c$ again from
the bottom to the top at a distance $x_{0}=1\times 10^{-4}\,\mathrm{m}$ to the
interface in the reflection region. The exponential attenuation of the
electric field in Fig. 3 is quite clear at the transmission region showing a
notably reduction of the radiation lobes in comparison with the ones at the
reflection region of Fig. 2.

Finally, we give two comments regarding the Figs. 2-3. First, we remark that
the typical VC cone always conserves its form without any modification. And
finally, we highlight that in comparison with the standard electrodynamics
case ($\Delta_{\Theta}=0$) the component $E_{y}$ is different from zero and
two orders of magnitude smaller than the other two components, which will
contribute to enhance the effect as Figs. 2b, 2e, 3b and 3e clearly show.

##  IV Angular Distribution of Energy

In this section, we will compute the angular distribution of radiation in the
upper and lower hemispheres focusing on the spherical wave contribution. To
this end, we will henceforth assume that $x_{0}\ll x$, therefore the inverse
distances $R_{0}$, $R_{1}\simeq r$ and the exponents $R_{0}\simeq r-x_{0}x/r$,
$R_{1}\simeq r+x_{0}x/r$ allowing us to rewrite the electric fields of Eqs.
(67) and (87) as follows

$\displaystyle\mathbf{E}_{1}(\mathbf{r};\omega)=\frac{\mathrm{i}q\omega\mu_{0}\mu_{1}(\omega)\mathcal{K}_{1}(\theta,\omega)}{4\pi
r}e^{\mathrm{i}k_{1}r}$
$\displaystyle\times\left\\{e^{-\mathrm{i}k_{1}x_{0}x/r}\left(\mathbf{e}_{z}-\cos\theta\mathbf{e}_{r,1}\right)\right.$
$\displaystyle+e^{\mathrm{i}k_{1}x_{0}x/r}\sin\theta\left[\cos\phi_{1}R_{\mathrm{TM,TM}}^{12}(\theta,\phi_{1},\Delta_{\Theta})\left(\mathbf{e}_{z}\times\mathbf{e}_{r,1}\right)\right.$
$\displaystyle\left.\left.-R_{\mathrm{TE,TM}}^{12}(\theta,\phi_{1},\Delta_{\Theta})\mathbf{e}_{\phi
1}\right]\right\\},$ (99)
$\displaystyle\mathbf{E}_{2}(\mathbf{r};\omega)=\frac{\mathrm{i}q\omega\mu_{0}\mu_{1}(\omega)k_{2}\mathcal{K}_{2}(\theta,\omega)}{4\pi
k_{1}r}e^{\mathrm{i}(k_{2}r+\xi x_{0})}$
$\displaystyle\times\sin\theta\left[\cos\phi_{2}T_{\mathrm{TM,TM}}^{12}(\theta,\phi_{2},\Delta_{\Theta})\left(\mathbf{e}_{z}\times\mathbf{e}_{r,2}\right)\right.$
$\displaystyle\left.+T_{\mathrm{TE,TM}}^{12}(\theta,\phi_{2},\Delta_{\Theta})\mathbf{e}_{\phi
2}\right],$ (100)

where
$\mathbf{e}_{r1,2}=(\sin\theta\cos\phi_{1,2},\sin\theta\sin\phi_{1,2},\cos\theta)$
and $\mathbf{e}_{\phi 1,2}=(-\sin\phi_{1,2},\cos\phi_{1,2},0)$. Here
$\theta\in[0,\pi]$ denote the zenith angle of the observer at both sides of
the interface. The azimuthal angle $\phi_{1,2}$ is defined for the upper
hemisphere in the interval $I_{\mathrm{UH}}=[0,\pi/2]\cup[3\pi/2,2\pi]$ and
$I_{\mathrm{LH}}=[\pi/2,3\pi/2]$ for the lower hemisphere. By means of these
basis we appreciate that the free space contribution oscillates throughout the
non-intuitive direction defined by
$\mathbf{e}_{z}-\cos\theta\mathbf{e}_{r,1}$. Then, the contributions related
to $R_{\mathrm{TM,TM}}^{12}$ and $T_{\mathrm{TM,TM}}^{12}$ oscillate in the
direction $\mathbf{e}_{z}\times\mathbf{e}_{r1,2}$ outside the interface
located at $x=0$. And, the contributions proportional to
$R_{\mathrm{TE,TM}}^{12}$ and $T_{\mathrm{TE,TM}}^{12}$ oscillate
perpendicular to the interface throughout the $\mathbf{e}_{\phi 1,2}$
direction.

We notice that the three vectors $\mathbf{E}_{1,2}$, $\mathbf{B}_{1,2}$, and
$\mathbf{e}_{r1,2}$ define a right-handed triad resulting in the Poynting
vector for the material medium 1 and 2 OJF-SYB . Then, the angular
distribution of the radiated energy per unit frequency in the interval
$z\in(-\zeta,\zeta)$ over the solid angle can be defined through the electric
field $\mathbf{E}_{1,2}$ in the following fashion Jackson ; Schwinger :

$\frac{d^{2}\mathcal{E}_{j}}{d\omega
d\Omega_{j}}=\sqrt{\frac{\varepsilon_{0}\varepsilon_{j}}{\mu_{0}\mu_{j}}}\frac{r^{2}}{\pi}\mathbf{E}_{j}^{*}(\mathbf{r};\omega)\cdot\mathbf{E}_{j}(\mathbf{r};\omega)\;,\\\
$ (101)

with $j=1,2$ and $d\Omega_{j}=\sin\theta d\theta d\phi_{j}$ being the solid
angle at the corresponding medium.

Before computing the two angular distributions, we discuss an important tool
that will simplify their calculations. In calculating the function
$\mathcal{K}_{1}$ (68) and $\mathcal{K}_{2}$ (88), if we consider the limit
$\zeta\gg v/\omega$ meaning effectively $\zeta\rightarrow\infty$, we encounter
expressions like $\sin(\zeta aN)/(aN)$ which behave as $\pi\delta(aN)$
Lighthill . We will take advantage of this delta-like behavior by setting all
terms involving $\cos\theta$ equal to $c/vn_{j}$.

###  IV.1 Upper hemisphere

After applying Eq. (101) to the spherical contribution of the electric field
(99), we find:

$\frac{d^{2}\mathcal{E}_{1}}{d\omega
d\Omega_{1}}=\sum_{j=1}^{5}\frac{d^{2}\mathcal{E}_{1}^{(j)}}{d\omega
d\Omega_{1}}\;,$ (102)

with leading-order terms in $1-1/\beta^{2}n_{1}^{2}$ being

$\frac{d^{2}\mathcal{E}_{1}^{(1)}}{d\omega
d\Omega_{1}}=\sqrt{\frac{\varepsilon_{0}\varepsilon_{1}}{\mu_{0}\mu_{1}}}\frac{q^{2}\omega^{2}\mu_{0}^{2}\mu^{2}_{1}\mathcal{K}_{1}^{2}(\theta,\omega)}{16\pi^{3}}\left(1-\frac{1}{\beta^{2}n_{1}^{2}}\right)\;,$
(103) $\displaystyle\frac{d^{2}\mathcal{E}_{1}^{(2)}}{d\omega d\Omega_{1}}$
$\displaystyle=$ $\displaystyle-2\frac{d^{2}\mathcal{E}_{1}^{(1)}}{d\omega
d\Omega_{1}}\mathrm{Re}\left[R_{\mathrm{TM,TM}}^{12}(\theta_{1}^{C},\phi_{1},\Delta_{\Theta})\right]\cos^{2}\phi_{1}$
(104)
$\displaystyle\times\cos\left(2k_{1}x_{0}\sin\theta_{1}^{C}\cos\phi_{1}\right)\;,$
$\displaystyle\frac{d^{2}\mathcal{E}_{1}^{(3)}}{d\omega
d\Omega_{1}}=\frac{2}{\beta n_{1}}\frac{d^{2}\mathcal{E}_{1}^{(1)}}{d\omega
d\Omega_{1}}\cos\phi_{1}\sin\phi_{1}$
$\displaystyle\times\mathrm{Re}\left[R_{\mathrm{TM,TM}}^{12}(\theta_{1}^{C},\phi_{1},\Delta_{\Theta})R_{\mathrm{TE,TM}}^{12\,*}(\theta_{1}^{C},\phi_{1},\Delta_{\Theta})\right],\quad\quad$
(105) $\frac{d^{2}\mathcal{E}_{1}^{(4)}}{d\omega
d\Omega_{1}}=\frac{d^{2}\mathcal{E}_{1}^{(1)}}{d\omega
d\Omega_{1}}\big{|}R_{\mathrm{TM,TM}}^{12}(\theta_{1}^{C},\phi_{1},\Delta_{\Theta})\big{|}^{2}\cos^{2}\phi_{1}\;,$
(106) $\frac{d^{2}\mathcal{E}_{1}^{(5)}}{d\omega
d\Omega_{1}}=\frac{d^{2}\mathcal{E}_{1}^{(1)}}{d\omega
d\Omega_{1}}\big{|}R_{\mathrm{TE,TM}}^{12}(\theta_{1}^{C},\phi_{1},\Delta_{\Theta})\big{|}^{2}\;,$
(107)

where $\beta=v/c$, $\sin\theta_{1}^{C}=\sqrt{1-1/(\beta^{2}n_{1}^{2})}$ and we
assumed frequency-independent values for
$\varepsilon_{1},\mu_{1},\varepsilon_{2}$ and $\mu_{2}$ in order to apply Eq.
(101), as well as in the rest of this article. Here the $\delta$-like behavior
of the function $\mathcal{K}_{1}$ described above was used implying an
evaluation of the involved Fresnel coefficients at the Cherenkov angle
$\theta_{1}^{C}$ (90). Also due to this application the characteristic factor
$1-1/\beta^{2}n_{1}^{2}$ of the VC radiation appears as a global factor
modulating the total angular distribution (102), which will always be
positive.

The important limit $\varepsilon_{1}=\varepsilon_{2}$, $\mu_{1}=\mu_{2}$ and
$\Delta_{\Theta}\neq 0$ describing a pure $\Delta_{\Theta}$ interface derived
directly from Eqs. (103)-(107) reads

$\displaystyle\frac{d^{2}\mathcal{E}_{1}}{d\omega
d\Omega_{1}}=\frac{d^{2}\mathcal{E}_{1}^{(1)}}{d\omega
d\Omega_{1}}\left[1-\left(R_{\mathrm{TM,TM}}^{11}\right)^{2}\cos^{2}\phi_{1}\right.$
(108)
$\displaystyle-2R_{\mathrm{TM,TM}}^{11}\cos^{2}\phi_{1}\cos\left(2k_{1}x_{0}\sin\theta_{1}^{C}\cos\phi_{1}\right)$
$\displaystyle\left.+\left(R_{\mathrm{TE,TM}}^{11}\right)^{2}+\frac{2}{\beta
n_{1}}R_{\mathrm{TM,TM}}^{11}R_{\mathrm{TE,TM}}^{11}\cos\phi_{1}\sin\phi_{1}\right].$

where we remark that the Fresnel coefficients are now real constants
determined by $\varepsilon_{1},\mu_{1}$ and $\Delta_{\Theta}$.

(a)

(b)

Figure 4:  Polar plots of the angular distribution of parallel VC radiation
along the VC cone in the reflection zone generated by a particle with
$v=0.95\,c$ parallel to the $z$ axis at a height $x_{0}=1\times
10^{-4}\,\mathrm{m}$ from the interface and with single frequency
$\omega=6.0450\times 10^{14}\,\,\mathrm{s}^{-1}$. A comparison between three
different scenarios is considered. The purple line corresponds to a
dielectric–dielectric interface ($\Delta_{\Theta}=0$) with different
permittivities $\varepsilon_{1}\neq\varepsilon_{2}$. The green solid line
represents the dielectric–TI interface ($\Delta_{\Theta}\neq 0$). And, the red
dotdashed line stands for the pure $\Delta_{\Theta}$ case when
$\varepsilon_{1}=\varepsilon_{2}$ and keeping the parameter
$\Delta_{\Theta}\neq 0$. The polar plots (a) compare these scenarios when the
topological insulator TlBiSe2 is at the lower layer. Meanwhile the polar plots
(b) compare these scenarios but with the magnetoelectric TbPO4 located at the
lower layer. Both materials are non magnetic ones. The radial axis indicates
the dimensionless angular distribution in the respective direction. The charge
moves along a vertical line that crosses the origin outside the sheet of
paper.

Some comments regarding this angular distribution are now in order. The
expression (102), as can be contrasted with its addends (103)-(107), has a
linear contribution in the topological parameter $\Delta_{\Theta}$, depends
explicitly on the particle’s distance $x_{0}$ to the interface and is neither
an even function of the angle $\phi_{1}$ nor an odd one, the last fact being a
consequence of the azimuthal symmetry breaking of the problem (recall Fig. 1).
The first term (103) is the free term corresponding to the standard VC
radiation. The second term (104) is an interference term due to the change of
phase gained from the electromagnetic field after reflection by the interface,
whose interference effects can be disregarded if
$\cos\left(2k_{1}x_{0}\sin\theta_{1}^{C}\cos\phi_{1}\right)\simeq 1$. The
third term (105) arises from the interference between the TM,TM and the TM,TE
polarizations and is linear in the topological parameter $\Delta_{\Theta}$.
This is evidence of the topological magnetoelectric effect which is absent in
the standard case ($\Delta_{\Theta}=0$). Nevertheless, if we apply the ideas
of fields generated by dynamical images of the work OJF-LFU-ORT —extensions of
the original ones developed in Refs. Qi Science ; Wilczek —this term is the
result of the interference between the electric field of an image magnetic
monopole of strength $g_{1}=qR_{\mathrm{TE,TM}}^{12}\sim-q\Delta_{\Theta}$ and
the one generated by an image electric charge with magnitude
$q^{\prime}=qR_{\mathrm{TM,TM}}^{12}\sim
q(\varepsilon_{2}-\varepsilon_{1}+\Delta_{\Theta}^{2})$. Then, the fourth term
(106) describes the interaction of the image electric charge with the
interface and analogously the last term (107) is related to the response of
the interface to the moving image magnetic monopole.

In addition, we observe that although the parallel VC radiation will be always
emitted in the typical VC cone (recall Figs. 2), it will not be emitted
uniformly because the functions involved in the terms (104)-(107) depend
explicitly on the azimuthal angle $\phi_{1}$, i.e. the parallel VC radiation
is not uniform over different generatrices of the VC cone Bolotovskii . Such
non uniformity is shown in the polar plots of Figs. 4a and 4b, where we plot
the angular distribution (102) divided by
$\mathcal{K}_{1}^{2}(\theta,\omega)\left(1-c^{2}/(v^{2}n_{1}^{2})\right)\sqrt{\varepsilon_{0}}q^{2}\omega^{2}\mu_{0}^{2}/(16\pi^{3}\sqrt{\mu_{0}})$
to appreciate the angular behavior, the interface lies on the line
$(\pi/2-3\pi/2)$ and the charge moves along a vertical line that crosses the
origin outside the sheet of paper with $v=0.95\,c$ at a height $x_{0}=1\times
10^{-4}\,\mathrm{m}$ from the interface. Fig. 4a illustrates this by comparing
three different interfaces: dielectric–dielectric ($\Delta_{\Theta}=0$),
dielectric–TI and pure $\Delta_{\Theta}$ ($\varepsilon_{1}=\varepsilon_{2}$
and $\Delta_{\Theta}\neq 0$). First, the dielectric–dielectric case having a
dielectric with $\varepsilon_{1}=1.2$, $\mu_{1}=1$ at the upper layer and
another dielectric with $\varepsilon_{2}=4$, $\mu_{2}=1$ at the lower layer
corresponds to the dashed purple line. Second, the green solid line represents
the dielectric–TI case with the same upper medium but the topological
insulator TlBiSe2 with $\varepsilon_{2}=4$, $\mu_{2}=1$ and
$\Delta_{\Theta}=11\alpha$ at the lower layer TlBiSe2 . And, the red dotdashed
line stands for the pure $\Delta_{\Theta}$ case given by Eq. (108) with the
upper and lower media having $\varepsilon_{1}=\varepsilon_{2}=1.2$,
$\mu_{1}=\mu_{2}=1$ and $\Delta_{\Theta}=11\alpha$, which is smaller than the
purple and green ones. In Fig. 4a we notice a big difference between the pure
$\Delta_{\Theta}$ case and the other two, where the former one presents an
almost uniform distribution of the radiation because the interference effects
are suppressed due to the small value of $\Delta_{\Theta}$ in contrast with
the other two angular distributions. This behavior reinforces the idea that
the topological parameter mimics a permittivity as analyzed in Refs. Crosse-
Fuchs-Buhmann ; OJF-SYB ; OJF-LFU  for other kinds of radiation.

In Fig. 4b we compare the same three cases but now for a material with a
stronger topological parameter. Fig. 4b has the same specifications of Fig.
4a. Here, the dielectric–dielectric case (dashed purple line) has a standard
dielectric medium with $\varepsilon_{1}=1.2$, $\mu_{1}=1$ at the upper layer
and another standard dielectric with $\varepsilon_{2}=3.4969$, $\mu_{2}=1$ at
the lower layer. Then, the dielectric–TI case (green solid line) is
constituted by the same upper medium and the magnetoelectric TbPO4 with
$\varepsilon_{2}=3.4969$, $\mu_{2}=1$ and $\Delta_{\Theta}=0.22$ at the lower
layer TbPO4 ; Rivera . Lastly, the pure $\Delta_{\Theta}$ case (red dotdashed
line) given by Eq. (108) is presented here for
$\varepsilon_{1}=\varepsilon_{2}=1.2$, $\mu_{1}=\mu_{2}=1$ and
$\Delta_{\Theta}=0.22$ being smaller greater than the purple and the green
lines. Interestingly, we appreciate an unexpected asymmetry with respect to
the polar axis between the angular distribution of the dielectric–dielectric
case and the one corresponding to the magnetoelectric TbPO4, because the
latter one results to be slightly bigger than the former one in the interval
$\phi_{1}\in[0,\pi/2]$ and vice versa in the interval
$\phi_{1}\in[3\pi/2,2\pi]$. This asymmetry is also present in Fig. 4a but is
difficult to appreciate there and is absent in the dielectric–dielectric case.
We can trace back its origin to the interference term given by Eq. (105) which
is sensible to a sign change in $\Delta_{\Theta}$ as certain 3D TIs present.
We observe that this asymmetry can be sharper here when the interference
pattern coming from the term (104) is suppressed, i.e. when
$\cos\left(2k_{1}x_{0}\sin\theta_{1}^{C}\cos\phi_{1}\right)\simeq 1$.
Regarding the pure $\Delta_{\Theta}$ case, we predominantly observe an
inhomogeneous distribution whose interference pattern becomes sharp when
$\phi_{1}\rightarrow 0$ due to the higher value of the topological parameter
$\Delta_{\Theta}$.

###  IV.2 Lower hemisphere

Here we apply Eq. (101) to the spherical contribution of the electric field
(100) and after assuming that $n_{1}<n_{2}$, we find:

$\frac{d^{2}\mathcal{E}_{2}}{d\omega
d\Omega_{2}}=\sum_{j=1}^{3}\frac{d^{2}\mathcal{E}_{2}^{(j)}}{d\omega
d\Omega_{2}}$ (109)

having the following leading-order terms in $1-1/\beta^{2}n_{2}^{2}$

$\displaystyle\frac{d^{2}\mathcal{E}_{2}^{(1)}}{d\omega
d\Omega_{2}}=2\sqrt{\frac{\varepsilon_{0}\varepsilon_{2}}{\mu_{0}\mu_{2}}}\frac{q^{2}\omega^{2}\mu_{0}^{2}\mu^{2}_{1}k_{2}^{2}\mathcal{K}_{2}^{2}(\theta,\omega)}{16\pi^{3}k_{1}^{2}\beta
n_{2}}\left(1-\frac{1}{\beta^{2}n_{2}^{2}}\right)$
$\displaystyle\times\mathrm{Re}\left[T_{\mathrm{TM,TM}}^{12}(\theta_{2}^{C},\phi_{2},\Delta_{\Theta})T_{\mathrm{TE,TM}}^{12\,*}(\theta_{2}^{C},\phi_{2},\Delta_{\Theta})\right]$
$\displaystyle\times e^{-2\Phi x_{0}}\cos\phi_{2}\sin\phi_{2}\;,$ (110)
$\displaystyle\frac{d^{2}\mathcal{E}_{2}^{(2)}}{d\omega d\Omega_{2}}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{\varepsilon_{0}\varepsilon_{2}}{\mu_{0}\mu_{2}}}\frac{q^{2}\omega^{2}\mu_{0}^{2}\mu^{2}_{1}k_{2}^{2}\mathcal{K}_{2}^{2}(\theta,\omega)}{16\pi^{3}k_{1}^{2}}\left(1-\frac{1}{\beta^{2}n_{2}^{2}}\right)$
(111)
$\displaystyle\times\big{|}T_{\mathrm{TM,TM}}^{12}(\theta_{2}^{C},\phi_{2},\Delta_{\Theta})\big{|}^{2}\cos^{2}\phi_{2}\,e^{-2\Phi
x_{0}},\;\;\;\quad$ $\displaystyle\frac{d^{2}\mathcal{E}_{2}^{(3)}}{d\omega
d\Omega_{2}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\varepsilon_{0}\varepsilon_{2}}{\mu_{0}\mu_{2}}}\frac{q^{2}\omega^{2}\mu_{0}^{2}\mu^{2}_{1}k_{2}^{2}\mathcal{K}_{2}^{2}(\theta,\omega)}{16\pi^{3}k_{1}^{2}}\left(1-\frac{1}{\beta^{2}n_{2}^{2}}\right)$
(112)
$\displaystyle\times\big{|}T_{\mathrm{TE,TM}}^{12}(\theta_{2}^{C},\phi_{2},\Delta_{\Theta})\big{|}^{2}e^{-2\Phi
x_{0}}\;,$

where we defined

$\Phi=\frac{\omega}{v}\sqrt{1-\beta^{2}n_{1}^{2}+\sin^{2}\phi_{2}\left(\beta^{2}n_{2}^{2}-1\right)}\;.$
(113)

To obtain Eqs.(110)-(112) we employed the same mathematical procedure and
assumptions of Sec. IV.1. Consequently, the characteristic factor
$1-1/\beta^{2}n_{2}^{2}$ of the VC radiation appears as a global factor
modulating the total angular distribution (109), which will always be
positive. Importantly results the global factor $e^{-2\Phi x_{0}}$ coming
directly from the factor $e^{\mathrm{i}\xi x_{0}}$ of the electric field
(100), which is a great difference in comparison with its upper hemisphere
analog and becomes a real decaying exponential due to the assumption
$n_{1}<n_{2}$ otherwise it simply cancels out with its conjugate in Eq. (101).
Thus, parallel VC radiation is highly suppressed in this region for distances
far away of the interface as the corresponding electric field plot shows in
Fig. 3. We will address this issue in the next section regarding the radiated
energy.

The relevant limit $\varepsilon_{1}=\varepsilon_{2}$, $\mu_{1}=\mu_{2}$ and
$\Delta_{\Theta}\neq 0$ describing a pure $\Delta_{\Theta}$ interface derived
directly from Eqs. (110)-(112) reads

$\displaystyle\frac{d^{2}\mathcal{E}_{2}}{d\omega d\Omega_{2}}$
$\displaystyle=\sqrt{\frac{\varepsilon_{0}\varepsilon_{2}}{\mu_{0}\mu_{2}}}\frac{q^{2}\omega^{2}\mu_{0}^{2}\mu^{2}_{2}\mathcal{K}_{2}^{2}(\theta,\omega)}{16\pi^{3}}\left(1-\frac{1}{\beta^{2}n_{2}^{2}}\right)$
(114)
$\displaystyle\times\left[\left(T_{\mathrm{TM,TM}}^{22}\right)^{2}\cos^{2}\phi_{2}+\left(T_{\mathrm{TE,TM}}^{22}\right)^{2}\right.$
$\displaystyle\left.+\frac{2}{\beta
n_{2}}T_{\mathrm{TM,TM}}^{22}T_{\mathrm{TE,TM}}^{22}\cos\phi_{2}\sin\phi_{2}\right]\;,$

where we emphasize that the Fresnel coefficients are real numbers and the
absence of the exponential factor.

(a)

(b)

Figure 5:  Polar plots of the angular distribution of parallel VC radiation
along the VC cone in the transmission zone generated by a particle with single
frequency $\omega=6.0450\times 10^{14}\,\,\mathrm{s}^{-1}$. A comparison
between three different scenarios is considered. The purple line corresponds
to a dielectric–dielectric interface ($\Delta_{\Theta}=0$) with different
permittivities $\varepsilon_{1}\neq\varepsilon_{2}$. The green solid line
represents the dielectric–TI interface ($\Delta_{\Theta}\neq 0$). And, the red
dotdashed line stands for the pure $\Delta_{\Theta}$ case when
$\varepsilon_{1}=\varepsilon_{2}$ and keeping the parameter
($\Delta_{\Theta}\neq 0$). The polar plots (a) compare these scenarios when
the topological insulator TlBiSe2 is at the lower layer with a charge velocity
$v=0.75\,c$. Meanwhile the polar plots (b) compare these scenarios but with
the magnetoelectric TbPO4 located at the lower layer and with a charge
velocity $v=0.9\,c$. Both materials are non magnetic ones. The radial axis
indicates the dimensionless angular distribution in the respective direction.
The charge moves along a vertical line that crosses the origin outside the
sheet of paper.

Notwithstanding that the angular distribution (109) has almost the same
characteristics as its upper hemisphere counterpart, we appreciate the absence
of the free term, since the particle moves at the other side of the interface.
By adapting the same arguments of the previous section, we can understand the
three terms (110)-(112) via the images interpretation but now the
corresponding image magnetic monopole has the strength
$g_{2}=qT_{\mathrm{TE,TM}}^{12}\sim q\Delta_{\Theta}$ and the magnitude of the
image electric charge is $q^{\prime\prime}=qT_{\mathrm{TM,TM}}^{12}\sim
q\varepsilon_{1}n_{2}/n_{1}$.

As in the reflective region, despite the parallel VC radiation always being
emitted in the typical VC cone (Recall Figs. 3), the non uniform radiation
emission over different generatrices of the VC cone is present because the
functions involved in the terms (110)-(112) depend explicitly on the azimuthal
angle $\phi_{2}$ Bolotovskii . We show this non-uniformity in the polar plots
of Figs. 5a and 5b, where we plot the angular distribution (109) divided by
$\mathcal{K}_{2}^{2}(\theta,\omega)e^{-2\Phi
x_{0}}\left(1-c^{2}/(v^{2}n_{2}^{2})\right)\sqrt{\varepsilon_{0}}q^{2}\omega^{2}\mu_{0}^{2}/(16\pi^{3}\sqrt{\mu_{0}})$
to appreciate the angular behavior. Figs. 5a and 5b have the same
specifications of Figs. 4a and 4b and compare the same three different
interfaces: dielectric–dielectric ($\Delta_{\Theta}=0$), dielectric–TI and
pure $\Delta_{\Theta}$ ($\varepsilon_{1}=\varepsilon_{2}$ and
$\Delta_{\Theta}\neq 0$), but they display two different velocities
$v=0.75\,c$ and $v=0.9\,c$ respectively. The pure $\Delta_{\Theta}$ case (red
dotdashed line) of Eq. (114) results to be clearly smaller than the purple and
green ones being the opposite situation of Fig. 4a and 4b. Here we notice that
the pure $\Delta_{\Theta}$ case presents an uniform distribution of the
radiation expected in a homogeneous medium in contrast with the other two
angular distributions.

Finally, we highlight the enhancement of the asymmetry with respect to the
polar axis between the angular distribution of the standard case and the one
corresponding to both the topological insulator TlBiSe2 and the
magnetoelectric TbPO4 in comparison to what is observed at the reflection
region in Fig. 4. Consistently, this asymmetry is due again to the
interference term given by Eq. (110) and is of course sensible to a sign
change in $\Delta_{\Theta}$ too.

##  V Radiated Energy

In this section we calculate the energy per unit frequency radiated by the
charge on its path from $-\zeta$ to $+\zeta$ at both sides of the interface.
To this end, we must integrate over the solid angle $\Omega_{j}$ to obtain
$d\mathcal{E}_{j}/d\omega$.

###  V.1 Upper hemisphere

By integrating the angular distribution of energy (102) over the solid angle
$\Omega_{1}$ in the upper hemisphere, we have

$\frac{d\mathcal{E}_{1}}{d\omega}=\sum_{j=1}^{5}\int_{\Omega_{1}}\frac{d^{2}\mathcal{E}_{1}^{(j)}}{d\omega
d\Omega_{1}}\;,$ (115)

which has the following five at order $1-1/\beta^{2}n_{1}^{2}$

$\frac{d\mathcal{E}_{1}^{(1)}}{d\omega}=\frac{q^{2}\omega\mu_{1}L}{8\pi\varepsilon_{0}c^{2}}\left(1-\frac{1}{\beta^{2}n_{1}^{2}}\right)\;,$
(116) $\displaystyle\frac{d\mathcal{E}_{1}^{(2)}}{d\omega}$ $\displaystyle=$
$\displaystyle-\frac{2}{\pi}\frac{d\mathcal{E}_{1}^{(1)}}{d\omega}\int_{I_{\mathrm{UH}}}d\phi_{1}\cos\left(2k_{1}x_{0}\sin\theta_{1}^{C}\cos\phi_{1}\right)$
(117)
$\displaystyle\times\mathrm{Re}\left[R_{\mathrm{TM,TM}}^{12}(\theta_{1}^{C},\phi_{1},\Delta_{\Theta})\right]\cos^{2}\phi_{1}\;,$
$\frac{d\mathcal{E}_{1}^{(3)}}{d\omega}=0\;,$ (118)
$\frac{d\mathcal{E}_{1}^{(4)}}{d\omega}=\frac{1}{\pi}\frac{d\mathcal{E}_{1}^{(1)}}{d\omega}\int_{I_{\mathrm{\mathrm{UH}}}}d\phi_{1}\cos^{2}\phi_{1}\big{|}R_{\mathrm{TM,TM}}^{12}(\theta_{1}^{C},\phi_{1},\Delta_{\Theta})\big{|}^{2},$
(119)
$\frac{d\mathcal{E}_{1}^{(5)}}{d\omega}=\frac{1}{\pi}\frac{d\mathcal{E}_{1}^{(1)}}{d\omega}\int_{I_{\mathrm{UH}}}d\phi_{1}\big{|}R_{\mathrm{TE,TM}}^{12}(\theta_{1}^{C},\phi_{1},\Delta_{\Theta})\big{|}^{2}\;,$
(120)

where we recalled that $I_{\mathrm{UH}}=[0,\pi/2]\cup[3\pi/2,2\pi]$ and
introduced the total length $L=2\zeta$ traveled by the particle, thus
obtaining that Eq. (116) is a half of the standard result Panofsky  because we
are restricted to the upper hemisphere. For full details, please see Appendix
D.

Now, we will carry out the integrals over $\phi_{1}$ for the interesting case
when the parallel VC radiation presents total internal reflection at the upper
medium, which is of practical interest for waveguides. To this end, we choose
a TI-vacuum interface with a nonmagnetical strong 3D TI with $\mu_{1}=1$ as
upper medium. As discussed in Sec. III, this will mean that all the radiation
is reflected and the VC radiation is absent in vacuum (the transmitted region
$x<0$). However, the integral of the second term of the angular distribution
given by Eq. (104) is very difficult to solve analytically but this difficulty
can be overcome by assuming
$\cos\left(2k_{1}x_{0}\sin\theta_{1}^{C}\cos\phi_{1}\right)\simeq 1$. After
transforming the interval of integration $I_{\mathrm{UH}}$ to $[-\pi/2,\pi/2]$
the resulting integrals at order $\Delta_{\Theta}^{2}$ are

$\displaystyle\frac{d\mathcal{E}_{1}^{(2)}}{d\omega}$ $\displaystyle=$
$\displaystyle\frac{q^{2}\omega
L}{8\pi\varepsilon_{0}c^{2}}\frac{1}{(\varepsilon_{1}+1)}\left\\{\left(1-\frac{1}{\beta^{2}n_{1}^{2}}\right)\mathcal{R}_{1}(\varepsilon_{1},\Delta_{\Theta})\right.$
(121)
$\displaystyle+\frac{2\varepsilon_{1}\mathcal{R}_{2}(\varepsilon_{1},\Delta_{\Theta})}{\varepsilon_{1}+1}\left[1-\frac{\sqrt{\varepsilon_{1}(\varepsilon_{1}-1)}}{\Gamma(\varepsilon_{1},\beta,\Delta_{\Theta})}\right]$
$\displaystyle\left.+\frac{\varepsilon_{1}(\varepsilon_{1}-1)}{\varepsilon_{1}-1-2\Delta_{\Theta}^{2}}\left[\frac{\sqrt{\varepsilon_{1}(\varepsilon_{1}-1)}}{\Gamma(\varepsilon_{1},\beta,\Delta_{\Theta})}-1\right]\right\\}\;,$
$\displaystyle\frac{d\mathcal{E}_{1}^{(4)}}{d\omega}$ $\displaystyle=$
$\displaystyle\frac{q^{2}\omega
L}{16\pi\varepsilon_{0}c^{2}}\frac{1}{(\varepsilon_{1}+1)}$ (122)
$\displaystyle\times\left\\{\left(1-\frac{1}{\beta^{2}n_{1}^{2}}\right)\left(\varepsilon_{1}-1\right)\mathcal{R}_{3}(\varepsilon_{1},\Delta_{\Theta})\right.$
$\displaystyle+\frac{2\left(\varepsilon_{1}-1\right)\mathcal{R}_{4}(\varepsilon_{1},\Delta_{\Theta})}{\varepsilon_{1}+1}\left[1-\frac{\sqrt{\varepsilon_{1}(\varepsilon_{1}-1)}}{\Gamma(\varepsilon_{1},\beta,\Delta_{\Theta})}\right]$
$\displaystyle+\left.\frac{2\varepsilon_{1}(\varepsilon_{1}-1)}{(\varepsilon_{1}-1-2\Delta_{\Theta}^{2})}\left[\frac{\sqrt{\varepsilon_{1}(\varepsilon_{1}-1)}}{\Gamma(\varepsilon_{1},\beta,\Delta_{\Theta})}-1\right]\right\\},\;\;\quad$

$\displaystyle\frac{d\mathcal{E}_{1}^{(5)}}{d\omega}$ $\displaystyle=$
$\displaystyle\frac{q^{2}\omega
L}{2\pi\varepsilon_{0}c^{2}}\left(1-\frac{1}{\beta^{2}n_{1}^{2}}\right)\frac{\Delta_{\Theta}^{2}}{(\varepsilon_{1}+1)(\varepsilon_{1}-1-2\Delta_{\Theta}^{2})}$
(123)
$\displaystyle\times\left[\frac{\sqrt{\varepsilon_{1}(\varepsilon_{1}-1)}}{\Gamma(\varepsilon_{1},\beta,\Delta_{\Theta})}-1\right],$

where we have defined

$\displaystyle\mathcal{R}_{1}(\varepsilon_{1},\Delta_{\Theta})=\frac{\varepsilon_{1}^{2}+1+2\Delta_{\Theta}^{2}}{\varepsilon_{1}-1-2\Delta_{\Theta}^{2}}\;,$
(124)
$\displaystyle\mathcal{R}_{2}(\varepsilon_{1},\Delta_{\Theta})=\frac{\varepsilon_{1}^{2}+1+2\Delta_{\Theta}^{2}}{\varepsilon_{1}-1-4\Delta_{\Theta}^{2}}\;,$
(125)
$\displaystyle\mathcal{R}_{3}(\varepsilon_{1},\Delta_{\Theta})=\frac{\varepsilon_{1}+1+2\Delta_{\Theta}^{2}}{\varepsilon_{1}-1-2\Delta_{\Theta}^{2}}\;,$
(126)
$\displaystyle\mathcal{R}_{4}(\varepsilon_{1},\Delta_{\Theta})=\frac{\varepsilon_{1}+1+2\Delta_{\Theta}^{2}}{\varepsilon_{1}-1-4\Delta_{\Theta}^{2}}\;,$
(127) $\displaystyle\Gamma(\varepsilon_{1},\beta,\Delta_{\Theta})=$
$\displaystyle\sqrt{\varepsilon_{1}(\varepsilon_{1}-1)-(\varepsilon_{1}+1)\left(1-\frac{1}{\beta^{2}n_{1}^{2}}\right)(\varepsilon_{1}-1-2\Delta_{\Theta}^{2})},$
(128)

with $\Gamma(\varepsilon_{1},\beta,\Delta_{\Theta})$ taken as a real function.
In this way, the radiated energy at the upper hemisphere (115) will consist of
the sum of (116) with $\mu_{1}=1$ and Eqs. (121)-(123), which will be
physically acceptable only when positive. We observe that due to the vanishing
interference term (118), the radiated energy is not sensitive to sign changes
on $\Delta_{\Theta}$.

###  V.2 Lower hemisphere

Analogously, through the method described immediately above, the energy per
unit frequency radiated by the charge in the lower hemisphere results

$\frac{d\mathcal{E}_{2}}{d\omega}=\sum_{j=1}^{3}\int_{\Omega_{2}}\frac{d^{2}\mathcal{E}_{2}^{(j)}}{d\omega
d\Omega_{2}}\;,$ (129)

where the adding terms at order $1-1/\beta^{2}n_{2}^{2}$ are the following

$\displaystyle\frac{d\mathcal{E}_{2}^{(1)}}{d\omega}$ $\displaystyle=$
$\displaystyle 0,\;$ (130)
$\displaystyle\frac{d\mathcal{E}_{2}^{(2)}}{d\omega}$ $\displaystyle=$
$\displaystyle\frac{q^{2}\omega\mu_{1}\varepsilon_{2}L}{8\pi^{2}\varepsilon_{0}\varepsilon_{1}c^{2}}\left(1-\frac{1}{\beta^{2}n_{2}^{2}}\right)\int_{\pi/2}^{3\pi/2}d\phi_{2}e^{-2\Phi
x_{0}}$ (131)
$\displaystyle\times\big{|}T_{\mathrm{TM,TM}}^{12}(\theta_{2}^{C},\phi_{2},\Delta_{\Theta})\big{|}^{2}\cos^{2}\phi_{2}\;,$
$\displaystyle\frac{d\mathcal{E}_{2}^{(3)}}{d\omega}$ $\displaystyle=$
$\displaystyle\frac{q^{2}\omega\mu_{1}\varepsilon_{2}L}{8\pi^{2}\varepsilon_{0}\varepsilon_{1}c^{2}}\left(1-\frac{1}{\beta^{2}n_{2}^{2}}\right)$
$\displaystyle\times\int_{\pi/2}^{3\pi/2}d\phi_{2}e^{-2\Phi
x_{0}}\big{|}T_{\mathrm{TE,TM}}^{12}(\theta_{2}^{C},\phi_{2},\Delta_{\Theta})\big{|}^{2}.$

The first integral corresponding to the interference term is zero because the
integrand is an odd function and after a change of variable one may shift the
interval of integration to $[-\pi/2,\pi/2]$. The integrals can be carried out
if we consider only the motion of a particle moving very close to the
interface. So, from the exponential factor and Eq. (113) we find that the
cutoff frequency at which the total radiated energy at the angle $\phi_{2}$
begins to be sharply suppressed is given by

$\omega_{\mathrm{cut}}=\frac{v}{2x_{0}\sqrt{1-n_{1}^{2}\beta^{2}+\sin^{2}\phi_{2}\left(n_{2}^{2}\beta^{2}-1\right)}}\;,$
(133)

which is exactly the same cutoff frequency reported by Bolotovskii Bolotovskii
but with $n_{1}\neq 1$. This coincidence and the absence of a dependency on
the topological parameter $\Delta_{\Theta}$ can be understood by recalling
that $\omega_{\mathrm{cut}}$ arises from $\Phi$ (113), which in turn
originates from the exponent $\xi$ of the electric field (100). Since the
latter is characterized only by the refractive indexes defined through bulk
properties, therefore $\omega_{\mathrm{cut}}$ must be unaffected by the
topological parameter $\Delta_{\Theta}$.

At this point, we will focus on the situation when the particle moves in
vacuum parallel to a nonmagnetical strong three-dimensional TI interface,
which is of practical interest for particle detectors. In this way, the VC
radiation in vacuum (the reflected region $x>0$) will be automatically ruled
out and all the posible radiation will be transmitted to the TI. So, by
assuming the condition (133) over the field’s frequency, we can set the
exponential factor equal to one and work out the integral over $\phi_{2}$ in
the two remaining expressions (131) and (LABEL:TRELH_3) obtaining at order
$\Delta_{\Theta}^{2}$

$\displaystyle\frac{d\mathcal{E}_{2}^{(2)}}{d\omega}=\frac{q^{2}\omega\varepsilon_{2}L}{2\pi\varepsilon_{0}c^{2}}\left\\{-\frac{1}{2(\varepsilon_{2}+1)}\right.$
(134)
$\displaystyle+\frac{1-\beta^{-2}}{(\varepsilon_{2}+1)(\varepsilon_{2}-1-2\Delta_{\Theta}^{2})}\left[1-\frac{\sqrt{\varepsilon_{2}(\varepsilon_{2}-1)}}{\Gamma(\varepsilon_{2},\beta,\Delta_{\Theta})}\right]$
$\displaystyle+\left.\frac{\beta^{2}\varepsilon_{2}^{3}\Gamma(\varepsilon_{2},\beta,\Delta_{\Theta})}{(\varepsilon_{2}+1)(\varepsilon_{2}-1-2\Delta_{\Theta}^{2})}\left[\frac{\varepsilon_{2}-1}{\Gamma(\varepsilon_{2},\beta,\Delta_{\Theta})}-\frac{\sqrt{\varepsilon_{2}-1}}{2\varepsilon_{2}\sqrt{\varepsilon_{2}}}\right]\right\\},$
$\displaystyle\frac{d\mathcal{E}_{2}^{(3)}}{d\omega}$ $\displaystyle=$
$\displaystyle\frac{q^{2}\omega\varepsilon_{2}L}{2\pi\varepsilon_{0}c^{2}}\left(1-\frac{1}{\beta^{2}\varepsilon_{2}}\right)\frac{\Delta_{\Theta}^{2}}{(\varepsilon_{2}+1)(\varepsilon_{2}-1-2\Delta_{\Theta}^{2})}$
(135)
$\displaystyle\times\left[\frac{\sqrt{\varepsilon_{2}(\varepsilon_{2}-1)}}{\Gamma(\varepsilon_{2},\beta,\Delta_{\Theta})}-1\right],$

where $\Gamma(\varepsilon_{2},\beta,\Delta_{\Theta})$ was defined on Eq. (128)
and also the interval of integration was shifted to $[-\pi/2,\pi/2]$. Again
$\Gamma(\varepsilon_{2},\beta,\Delta_{\Theta})$ must be real in expressions
(134) and (135). By adding these terms, the radiated energy at the lower
hemisphere (129) can be found, which will be physically consistent when
positive. And, as found in the radiated energy at the upper hemisphere, this
radiated energy is insensible to changes on the topological parameter
$\Delta_{\Theta}$ because of the vanishing interference term (130).

Inspecting Eqs. (121)-(123) for the radiated energy in the upper hemisphere
and the corresponding ones for the lower hemisphere given by Eqs. (134) and
(135), we appreciate an intricate algebraic dependency on the velocity as
Bolotovskii’s single result Bolotovskii . Despite the similarities, they do
not coincide in the dielectric–dielectric case ($\Delta_{\Theta}=0$) because
Bolotovskii’s expression is valid in all space resulting impossible to
distinguish correctly between the reflection and transmissive contributions of
the electric field. In contrast, ours is naturally split into reflection and
transmissive contributions due to the Green’s function method. The final
comment of this section is to stress that this algebraic dependency on the
velocity is completely different in comparison with the logarithmic one that
we obtained also by means of the Green’s function method for transition
radiation (perpendicular motion towards the interface) near this kind of
materials OJF-SYB .

###  V.3 Pure $\Delta_{\Theta}$ interface

It is worth to study the radiated energy for the pure $\Delta_{\Theta}$ case,
i.e. an interface with equal permittivities $\varepsilon_{1}=\varepsilon_{2}$,
same permeabilities $\mu_{1}=\mu_{2}$ but $\Delta_{\Theta}\neq 0$.
Consequently, the involved reflection and transmission coefficients do not
longer depend on the angles and become constants. Thus, after solving the
integrals (117)-(120) including interference effects, the radiated energy at
the upper hemisphere (115) results in

$\displaystyle\frac{d\mathcal{E}_{1}}{d\omega}=\frac{q^{2}\omega\mu_{1}L}{8\pi\varepsilon_{0}c^{2}}\left(1-\frac{1}{\beta^{2}n_{1}^{2}}\right)$
$\displaystyle\times\left\\{1+\frac{1}{2}\left(R_{\mathrm{TM,TM}}^{11}\right)^{2}+\left(R_{\mathrm{TE,TM}}^{11}\right)^{2}-2R_{\mathrm{TM,TM}}^{11}\right.$
$\displaystyle\left.\times\left[\frac{J_{1}\left(2k_{1}x_{0}\sin\theta_{1}^{C}\right)}{2k_{1}x_{0}\sin\theta_{1}^{C}}-J_{2}\left(2k_{1}x_{0}\sin\theta_{1}^{C}\right)\right]\right\\},$
(136)

where $J_{1}$ and $J_{2}$ are Bessel functions. If we consider that the
topological parameter $\Delta_{\Theta}$ is smaller than 1, we may approximate
Eq. (136) at order $\Delta_{\Theta}^{2}$ as follows

$\displaystyle\frac{d\mathcal{E}_{1}}{d\omega}=\frac{q^{2}\omega\mu_{1}L}{8\pi\varepsilon_{0}c^{2}}\left(1-\frac{1}{\beta^{2}n_{1}^{2}}\right)$
$\displaystyle\times\left\\{1+\frac{\Delta_{\Theta}^{2}}{2(\Delta_{\Theta}^{2}+2\varepsilon\mu^{3})}-\frac{2\Delta_{\Theta}^{2}}{\Delta_{\Theta}^{2}+4\varepsilon\mu^{3}}\right.$
$\displaystyle\left.\times\left[\frac{J_{1}\left(2k_{1}x_{0}\sin\theta_{1}^{C}\right)}{2k_{1}x_{0}\sin\theta_{1}^{C}}-J_{2}\left(2k_{1}x_{0}\sin\theta_{1}^{C}\right)\right]\right\\},\quad$
(137)

where we observe how the interference is encoded in the term given by the
Bessel functions and is purely modulated by the topological parameter although
the permittivities and permeabilities of both media are the same. Let us
complete the analysis of this radiated energy by studying its limit for
distances very close to the interface $x_{0}\rightarrow 0$,

$\frac{d\mathcal{E}_{1}}{d\omega}=\frac{q^{2}\omega\mu_{1}L}{8\pi\varepsilon_{0}c^{2}}\left(1-\frac{1}{\beta^{2}n_{1}^{2}}\right)=\frac{d\mathcal{E}_{1}^{(1)}}{d\omega}\;,$
(138)

where terms of order higher than $\Delta_{\Theta}^{2}$ were consistently
discarded and we observe that this result coincides with Eq. (116). A quick
comparison between Eqs. (137) and (138) shows that only non-negligible
distances to the interface $x_{0}$ lead to new effects.

On the other hand, after solving the integrals (134) and (135), we write the
radiated energy at the lower hemisphere (129) in the following fashion

$\displaystyle\frac{d\mathcal{E}_{2}}{d\omega}$ $\displaystyle=$
$\displaystyle\frac{q^{2}\omega\mu_{1}L}{8\pi^{2}\varepsilon_{0}c^{2}}\left(1-\frac{1}{\beta^{2}n_{1}^{2}}\right)$
(139)
$\displaystyle\times\left[\frac{1}{2}\left(T_{\mathrm{TM,TM}}^{11}\right)^{2}+\left(T_{\mathrm{TE,TM}}^{11}\right)^{2}\right]\;,$
$\displaystyle\simeq$
$\displaystyle\frac{1}{2}\frac{q^{2}\omega\mu_{1}L}{8\pi^{2}\varepsilon_{0}c^{2}}\left(1-\frac{1}{\beta^{2}n_{1}^{2}}\right)=\frac{1}{2}\frac{d\mathcal{E}_{1}}{d\omega}\;,$

where terms of order $\Delta_{\Theta}^{4}$ have been neglected in the second
step. We appreciate that Eq. (139) is almost the remaining half of the well-
known result of the standard VC radiation energy Panofsky , which is
reasonable due to our integration over the other half of the whole space.
Nonetheless, there is a missing half of the energy at the lower hemisphere,
because in the limit $\Delta_{\Theta}=0$, the sum of Eqs. (136) and (139) must
reproduce the standard VC radiated energy Panofsky . At this stage, we do not
know whether the missing half is converted into non propagating radiation or
there is a fundamental reason for this result. Notwithstanding this
unexplained result, the non-dependency on the topological parameter of the
radiated energies (136) and (139) is notable showing that for a traveling
particle quite close to a pure $\Delta_{\Theta}$ interface its VC radiation
results unaltered. This resembles the situation found in the same situation
with the perpendicular motion of the work OJF-LFU-ORT .

##  VI Retarding force

(a)

(b)

Figure 6:  Plots of the retarding force $F_{z}$ as function of $\beta$ due to
parallel VC radiation in the reflection zone experienced by a particle with
single frequency $\omega=6.0450\times 10^{14}\,\,\mathrm{s}^{-1}$. (a)
Retarding force for a dielectric–dielectric interface ($\Delta_{\Theta}=0$)
having $\varepsilon_{1}=1.2$ at the upper layer and the topological insulator
TlBiSe2 with $\varepsilon_{2}=4$ and $\Delta_{\Theta}=\alpha$ at the lower
layer for different values of the height $x_{0}$ to the interface. (b)
Comparison between two scenarios for the retarding force at the same height
$x_{0}=1\times 10^{-4}\,\mathrm{m}$. The purple line corresponds to a
dielectric–dielectric interface ($\Delta_{\Theta}=0$) whose permittivities are
$\varepsilon_{1}=1.2$ and $\varepsilon_{2}=3.4969$. The green solid line
represents the same quantity but now with the dielectric $\varepsilon_{1}=1.2$
at the upper layer and the magnetoelectric TbPO4 with $\varepsilon_{2}=3.4969$
and $\Delta_{\Theta}=0.22$ located at the lower layer. The scale is in
arbitrary dimensions and all materials are non magnetic.

Another quantity that is of interest to analyze is the retarding force
experienced by the charged particle when it moves parallel to the interface
between a dielectric and a nonmagnetic TI. The retarding force, which is
calculated through the energy loss per unit length Bolotovskii ; Schwinger
reads

$F_{z}=-\frac{d\mathcal{E}_{1}}{dL}=\sum_{j=1}^{5}F_{z}^{(j)}=-\sum_{j=1}^{5}\frac{d\mathcal{E}_{1}^{(j)}}{dL}\;.$
(140)

Due to practical reasons, we will focus on the reflection part of the force
which is in principle accessible to measurement. To do so, we first integrate
Eq. (102) or equivalently Eqs. (103)-(107) over $\theta$ in this context. Then
we differentiate it with respect to the total length $L$ and solve the
integral over $\omega\in[0,\omega_{\mathrm{cut}}]$, where
$\omega_{\mathrm{cut}}$ is given by Eq. (133). To carry out the frequency
integral, we estimate a cutoff frequency by allowing wavelengths for which the
exponential factor appearing in Eq. (113) can be set equal to unity, resulting
in $\omega_{\mathrm{cut}}\approx
c/2x_{0}\sqrt{\varepsilon_{2}-\varepsilon_{1}}$ Bolotovskii . The last step is
to integrate over $\phi_{1}$ obtaining finally $d\mathcal{E}_{1}/dL$. So,
after applying this procedure the retarding force is

$F_{z}^{(1)}=-\frac{q^{2}}{64\pi\varepsilon_{0}x_{0}^{2}(\varepsilon_{2}-\varepsilon_{1})}\left(1-\frac{1}{\beta^{2}n_{1}^{2}}\right)\;,$
(141) $\displaystyle F_{z}^{(2)}$ $\displaystyle=$
$\displaystyle+\frac{q^{2}}{32\pi^{2}\varepsilon_{0}x_{0}^{2}(\varepsilon_{2}-\varepsilon_{1})}\left(1-\frac{1}{\beta^{2}n_{1}^{2}}\right)$
(142)
$\displaystyle\times\int_{-\pi/2}^{\pi/2}d\phi_{1}\cos\left(2k_{1}x_{0}\sin\theta_{1}^{C}\cos\phi_{1}\right)$
$\displaystyle\times\mathrm{Re}\left[R_{\mathrm{TM,TM}}^{12}(\theta_{1}^{C},\phi_{1},\Delta_{\Theta})\right]\cos^{2}\phi_{1}\;,$
$F_{z}^{(3)}=0\;,$ (143) $\displaystyle F_{z}^{(4)}$ $\displaystyle=$
$\displaystyle-\frac{q^{2}}{64\pi^{2}\varepsilon_{0}x_{0}^{2}(\varepsilon_{2}-\varepsilon_{1})}\left(1-\frac{1}{\beta^{2}n_{1}^{2}}\right)$
$\displaystyle\times\int_{-\pi/2}^{\pi/2}d\phi_{1}\cos^{2}\phi_{1}\big{|}R_{\mathrm{TM,TM}}^{12}(\theta_{1}^{C},\phi_{1},\Delta_{\Theta})\big{|}^{2}\;,$
$\displaystyle F_{z}^{(5)}$ $\displaystyle=$
$\displaystyle-\frac{q^{2}}{64\pi^{2}\varepsilon_{0}x_{0}^{2}(\varepsilon_{2}-\varepsilon_{1})}\left(1-\frac{1}{\beta^{2}n_{1}^{2}}\right)$
(145)
$\displaystyle\times\int_{-\pi/2}^{\pi/2}d\phi_{1}\big{|}R_{\mathrm{TE,TM}}^{12}(\theta_{1}^{C},\phi_{1},\Delta_{\Theta})\big{|}^{2}\;,$

where again the interval of integration over $\phi_{1}$ was changed from
$I_{\mathrm{UH}}$ to $[-\pi/2,\pi/2]$ as in Sec. V.1 and Appendix D. Although
the last two integrals can be solved analytically, that of Eq. (143) involving
the interference term cannot be analytically solved. For this reason, we opted
to solve numerically the angular integrals and provide plots of the retarding
force as shown in Fig. 6.

From Eqs. (141)-(LABEL:Force4), we observe that the retarding force has a
Coulombian behavior upon the interference effects due to Eq. (143), it is an
even function in $\Delta_{\Theta}$ and the characteristic factor
$1-1/\beta^{2}n_{1}^{2}$ of the VC radiation appears as a global factor.

Fig. 6 displays the retarding force $F_{z}$ as function of $\beta$ and shows
the repulsive character of the force meaning that it is opposed to the
movement of the charge as Bolotovskii found for the dielectric–dielectric
interface Bolotovskii . In Fig. 6a we plot the retarding force at constant
$\Delta_{\Theta}$ and different heights $x_{0}$ to a dielectric–TI interface
whose dielectric has $\varepsilon_{1}=1.2$ and $\Delta_{\Theta}=0$ and TlBiSe2
as its TI with $\varepsilon_{2}=4$ and $\Delta_{\Theta}=\alpha$ TlBiSe2 . This
plot shows an increase of the force magnitude for small heights $x_{0}$.
Meanwhile in Fig. 6b we compare the retarding force at constant height
$x_{0}=1\times 10^{-4}\,\mathrm{m}$ from a dielectric–dielectric interface and
a dielectric–TI one. The dielectric–dielectric case (dashed purple line) has a
dielectric with $\varepsilon_{1}=1.2$ and $\Delta_{\Theta}=0$ in its upper
layer and a dielectric with $\varepsilon_{2}=3.4969$ in the lower one. Aiming
to illustrate the differences of an increasing topological parameter, the
dielectric–TI case (green line) has the dielectric $\varepsilon_{1}=1.2$ at
the upper layer and the magnetoelectric TbPO4 with $\varepsilon_{2}=3.4969$
and $\Delta_{\Theta}=0.22$ at the lower one TbPO4 ; Rivera . Here we clearly
appreciate an enhancement in the retarding force in the ultra-relativistic
regime due to the magnetoelectric TbPO4 resembling the behavior found for
transition radiation in the same regime OJF-SYB .

##  VII Conclusions

We have analyzed the parallel Vavilov-Čerenkov radiation emitted by a charged
particle propagating with constant velocity $v$ along the $z$ axis parallel to
the interface between two generic magnetoelectric media with special emphasis
on TIs through the Green’s function method. By means of the far-field
approximation together with the steepest descent method, we were able to
obtain analytical expressions for the electromagnetic field, which is due to a
superposition of spherical waves and lateral waves with contributions of both
kind associated to a purely topological origin.

In the radiation zone, we study the electric field and the angular
distribution of the radiation, whose behaviors for different kinds of
interface were illustrated in Figs. 2-5 respectively. The analysis of these
quantities show that the main characteristics of the parallel Vavilov-Čerenkov
radiation modified by strong three-dimensional TIs are the following. (i) The
Vavilov-Čerenkov angle remains the same and only varies depending on which
side of interface the observer is located as described first by the Ref.
Bolotovskii  for a dielectric–dielectric interface. (ii) Although this
radiation is also confined to the Vavilov-Čerenkov cone, it is emitted in an
inhomogeneous and asymmetric way. Remarkably, the asymmetry of this radiation
comes from a term proportional to the topological parameter $\Delta_{\Theta}$,
which is sensitive to the sign of it and could serve to measure it indirectly.
(iii) As in standard electrodynamics the height $x_{0}$ of the source to the
interface results crucial to enhance (suppress) the interference effects in
the reflected region and the transmitted radiation.

Our expressions at order $\Delta_{\Theta}^{2}$ for the radiated energy of this
radiation at both sides of the interface show an algebraic dependence on the
velocity but are not sensitive to changes in the sign of the topological
parameter $\Delta_{\Theta}$ due to the angular integration. However, the
topological parameter contributes with extra terms that are absent in a
dielectric–dielectric interface.

General expressions for the reaction force experienced by the charge were also
derived and after numerical integration, we found that it is repulsive and has
a Coulomb-like behavior proportional to the squared inverse distance of the
charge to the interface as Fig. 6a shows. Despite the lack of sensitiveness to
changes in the sign of the topological parameter $\Delta_{\Theta}$ by the
retarding force, it receives a significant increase in the ultra-relativistic
regime due to the additional terms arising from the topological parameter as
can be appreciated in Fig. 6b. It is hoped that this classical treatment of
the retarding force provides insights and motivates further studies on its
quantum analog, the quantum friction QF-Scheel-Buhmann ; Pieplow-Henkel .

Finally, we address the question of measuring these effects. In practice,
Čerenkov detectors actually measure the number of photons radiated per unit
length and per unit frequency, which can be obtained from Eqs. (115) and
(129). To probe this quantity, it is preferred an experimental setup with an
interface having $\varepsilon_{1}\neq\varepsilon_{2}$ otherwise it will result
difficult to distinguish a new effect as our Eqs. (136) and (137) for the pure
$\Delta_{\Theta}$ case show.

We note that these findings provide an indirect measurement of the topological
parameter $\Delta_{\Theta}$, aligning with propositions in existing literature
Wu ; Dziom ; Okada . Notably, similar effects can manifest in axion insulators
AXIs , sharing structural characteristics with topological insulators such as
gapped bulk and surface states, albeit with topological properties protected
by inversion symmetry rather than time-reversal symmetry. It is hoped that the
closed form expressions for the reflected parallel Vavilov-Čerenkov radiation
will be beneficial for developing waveguides in magnetoelectric media.
Moreover, this study holds significance for ongoing investigations into dark
matter detection, where axions emerge as a promising candidate Sikivie . In
this pursuit, condensed matter physics offers novel avenues through TIs Nenno
et al , antiferromagnetically doped TIs Marsh et al ; Jan Schuette , or
multiferroics Roising et al , leveraging the magnetoelectric effect which
underpins our results.

##  VIII Acknowledgments

O. J. F. has been supported by the postdoctoral fellowship CONACYT-800966 and
by the German Research Foundation (DFG, Project No. 328961117-SFB 1319 ELCH).

##  Appendix A Green’s Function components relevant for the electric field

In this appendix, we provide the necessary details of the Green’s tensor
$\mathbb{G}(\mathbf{r},\mathbf{r}^{\prime};\omega)$ required to obtain the
electric field via Eq. (9) for this configuration. First, the free space part
$\mathbb{G}^{(0)}(\mathbf{r},\mathbf{r}^{\prime};\omega)$ of the Green’s
tensor, proportional to $\mathds{1}$, is given by

$\displaystyle\mathbb{G}^{(0)}(\mathbf{r},\mathbf{r}^{\prime};\omega)$
$\displaystyle=$ $\displaystyle\mu(\omega)\frac{e^{\mathrm{i}kR}}{4\pi
R}\left[\left(1+\frac{\mathrm{i}kR-1}{k^{2}R^{2}}\right)\mathds{1}\right.$
(146)
$\displaystyle\left.+\frac{3-3\mathrm{i}kR-k^{2}R^{2}}{k^{2}R^{2}}\frac{\mathbf{R}\otimes\mathbf{R}}{R^{2}}\right]\;,$

where $\mathbf{R}=\mathbf{r}-\mathbf{r}^{\prime}$.

Then the reflective part of the Green’s tensor
$\mathbb{G}^{(1)}_{\mathbf{r},\mathbf{r}^{\prime}>0}$ of Eq. (11) can be
rewritten as

$\mathbb{G}^{(1)\;ij}(\mathbf{r},\mathbf{r}^{\prime};\omega)=\int\frac{d^{2}\mathbf{k}_{\parallel}}{(2\pi)^{2}}e^{\mathrm{i}\mathbf{k}_{\parallel}\cdot\mathbf{R}_{\parallel}}R^{ij}\left(x,x^{\prime};\mathbf{k}_{\parallel},\omega\right)\;,$
(147)

with $\mathbf{R}_{\parallel}=(y-y^{\prime},z-z^{\prime})$,
$\mathbf{k}_{\parallel}=(k_{y},k_{z})$ is the transversal wavevector (parallel
to the interface), and whose only required components in the present work are

$\displaystyle
R^{\,xz}\left(x,x^{\prime};\mathbf{k}_{\parallel},\omega\right)=\frac{\mathrm{i}\mu(\omega)}{2k_{x}}e^{\mathrm{i}k_{x}(|x|+|x^{\prime}|)}$
$\displaystyle\times\left[\mathrm{sgn}(x^{\prime})\frac{k_{z}k_{x}}{k^{2}}R_{\mathrm{TM,TM}}+\frac{k_{y}}{k}R_{\mathrm{TM,TE}}\right]\;,$
(148) $\displaystyle
R^{\,yz}\left(x,x^{\prime};\mathbf{k}_{\parallel},\omega\right)=\frac{\mathrm{i}\mu(\omega)}{2k_{x}}e^{\mathrm{i}k_{x}(|x|+|x^{\prime}|)}$
$\displaystyle\times\left[-\frac{k_{z}k_{y}}{k_{\parallel}^{2}}R_{\mathrm{TE,TE}}-\frac{k_{z}k_{y}k_{x}^{2}}{k_{\parallel}^{2}k^{2}}R_{\mathrm{TM,TM}}\right.$
$\displaystyle\left.-\mathrm{sgn}(x^{\prime})\frac{k_{x}}{k^{2}_{\parallel}k}\left(k_{z}^{2}R_{\mathrm{TE,TM}}+k_{y}^{2}R_{\mathrm{TM,TE}}\right)\right]\;,$
(149) $\displaystyle
R^{\,zz}\left(x,x^{\prime};\mathbf{k}_{\parallel},\omega\right)$
$\displaystyle=\frac{\mathrm{i}\mu(\omega)}{2k_{x}}e^{\mathrm{i}k_{x}(|x|+|x^{\prime}|)}\left[\frac{k_{y}^{2}}{k_{\parallel}^{2}}R_{\mathrm{TE,TE}}-\frac{k_{z}^{2}k_{x}^{2}}{k_{\parallel}^{2}k^{2}}R_{\mathrm{TM,TM}}\right.$
$\displaystyle\left.+\mathrm{sgn}(x^{\prime})\frac{k_{x}}{k^{2}_{\parallel}k}\left(k_{z}k_{y}R_{\mathrm{TE,TM}}-k_{z}k_{y}R_{\mathrm{TM,TE}}\right)\right],$
(150)

where $\mu(\omega)$ is the permeability of the upper medium, $\mathrm{sgn}$
stands for the sign function and we recall that
$k_{x}=\sqrt{k^{2}-\mathbf{k}_{\parallel}^{2}}$.

Similarly, the transmissive part of the Green’s function
$\mathbb{G}^{(1)}_{\mathbf{r},\mathbf{r}^{\prime}<0}$ of Eq. (11) can be
rewritten as

$\mathbb{G}^{(1)\;ij}(\mathbf{r},\mathbf{r}^{\prime};\omega)=\int\frac{d^{2}\mathbf{k}_{\parallel}}{(2\pi)^{2}}e^{\mathrm{i}\mathbf{k}_{\parallel}\cdot\mathbf{R}_{\parallel}}T^{ij}\left(x,x^{\prime};\mathbf{k}_{\parallel},\omega\right)\;,$
(151)

whose only required components in the present work are

$\displaystyle
T^{\,xz}\left(x,x^{\prime};\mathbf{k}_{\parallel},\omega\right)=\frac{\mathrm{i}\mu^{\prime}(\omega)}{2k_{x^{\prime}}}e^{\mathrm{i}k_{x}|x|+ik_{x^{\prime}}|x^{\prime}|}$
$\displaystyle\times\left[\mathrm{sgn}(x^{\prime})\frac{k_{z}k_{x^{\prime}}}{kk^{\prime}}T_{\mathrm{TM,TM}}+\frac{k_{y}}{k}T_{\mathrm{TM,TE}}\right]\;,$
(152) $\displaystyle
T^{\,yz}\left(x,x^{\prime};\mathbf{k}_{\parallel},\omega\right)=\frac{\mathrm{i}\mu^{\prime}(\omega)}{2k_{x^{\prime}}}e^{\mathrm{i}k_{x}|x|+\mathrm{i}k_{x^{\prime}}|x^{\prime}|}$
$\displaystyle\times\left[-\frac{k_{z}k_{y}}{k_{\parallel}^{2}}T_{\mathrm{TE,TE}}+\frac{k_{z}k_{y}k_{x}k_{x^{\prime}}}{k_{\parallel}^{2}kk^{\prime}}T_{\mathrm{TM,TM}}\right.$
$\displaystyle\left.-\frac{\mathrm{sgn}(x^{\prime})}{k^{2}_{\parallel}}\left(k_{z}^{2}\frac{k_{x^{\prime}}}{k^{\prime}}T_{\mathrm{TE,TM}}-k_{y}^{2}\frac{k_{x}}{k}T_{\mathrm{TM,TE}}\right)\right]\;,$
$\displaystyle
T^{\,zz}\left(x,x^{\prime};\mathbf{k}_{\parallel},\omega\right)$
$\displaystyle=\frac{\mathrm{i}\mu^{\prime}(\omega)}{2k_{x^{\prime}}}e^{\mathrm{i}k_{x}|x|+\mathrm{i}k_{x^{\prime}}|x^{\prime}|}\left[\frac{k_{y}^{2}}{k_{\parallel}^{2}}T_{\mathrm{TE,TE}}+\frac{k_{z}^{2}k_{x}k_{x^{\prime}}}{k_{\parallel}^{2}kk^{{}^{\prime}}}T_{\mathrm{TM,TM}}\right.$
$\displaystyle\left.+\mathrm{sgn}(x^{\prime})\frac{k_{z}k_{y}}{k^{2}_{\parallel}}\left(\frac{k_{x^{\prime}}}{k^{\prime}}T_{\mathrm{TE,TM}}+\frac{k_{x}}{k}T_{\mathrm{TM,TE}}\right)\right],$
(154)

where $k_{x^{\prime}}=\sqrt{k^{{}^{\prime}\,2}-\mathbf{k}_{\parallel}^{2}}$,
and $k^{\prime}$ and $\mu^{\,\prime}(\omega)$ denote the wavenumber and the
permeability of the source medium.

Once the labels for the upper and lower media are designed. Through Eqs.
(146), (147) and (151) we obtain the four possible cases for a two layer
configuration: observer and source at the upper medium, observer and source at
the lower medium, observer at the upper medium and source at the lower one,
and vice versa.

##  Appendix B Approximating the required integrals for the electric field

In this Appendix, we utilize the steepest descent method to derive the far-
field approximation of the $k_{\parallel}$ integrals pertaining to both
reflective (20)-(22) and transmissive electric fields (29)-(31). For
conciseness, we will solely illustrate the calculations for the $x$ component,
as expressed in Eq. (20). Similar procedures are employed to obtain the
remaining relevant components. Our approach closely adheres to Ref. Chew ,
which furnishes a meticulous explanation of the general procedure for handling
the steepest descent method concerning this class of integrals.

The three $k_{\parallel}$ integrals that need to be determined in
$E^{\,x}_{1}(\mathbf{r};\omega)$ are

$\mathcal{I}_{0}=\int_{-\zeta}^{\zeta}dz^{\prime}\frac{e^{\mathrm{i}k_{1}R}}{4\pi
R}\left[-\frac{(x-x_{0})(z-z^{\prime})}{R^{2}}\right]e^{\mathrm{i}\frac{\omega
z^{\prime}}{v}}\;,$ (155)
$\mathcal{I}_{1}=\int_{0}^{\infty}dk_{\parallel}k_{\parallel}R_{\mathrm{TM,TM}}^{12}(k_{\parallel})J_{0}(k_{\parallel}R_{\parallel})e^{\mathrm{i}k_{x,1}(x+x_{0})}\;,$
(156)
$\mathcal{I}_{2}=\int_{0}^{\infty}\frac{dk_{\parallel}k_{\parallel}}{k_{x,1}}R_{\mathrm{TM,TE}}^{12}(k_{\parallel})J_{0}(k_{\parallel}R_{\parallel})e^{\mathrm{i}k_{x,1}(x+x_{0})}\;,$
(157)

where $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ were defined in Eqs. (23) and
(24).

Let us begin with $\mathcal{I}_{0}$. Recalling that
$R_{\parallel}=\sqrt{y^{2}+(z-z^{\prime})^{2}}$,
$R^{2}=(x-x_{0})^{2}+R_{\parallel}^{2}$ and by considering the coordinate
conditions

$x\pm x_{0}\rightarrow\infty\;,\;\;y\rightarrow\infty\;,\;\;z^{\prime}\ll
z\rightarrow\infty\;,$ (158)

we write $R$ in the far-field approximation as $R\simeq
R_{0}-zz^{\prime}/R_{0}$ with $R_{0}=\sqrt{(x-x_{0})^{2}+\rho^{2}}$ and
$\rho=\sqrt{y^{2}+z^{2}}$. Therefore, we find

$\mathcal{I}_{0}=-\frac{e^{\mathrm{i}k_{1}R_{0}}}{4\pi
R_{0}}\frac{(x-x_{0})z}{R_{0}^{2}}\mathcal{K}_{1}(z/R_{0},\omega)\;,$ (159)

where the function (68) was used. This expression coincides with the $x$
component of the free-space contribution of the field (67).

Now, our focus shifts to computing the integral $\mathcal{I}_{1}$. First, we
go to the complex plane by expressing the Bessel function $J_{0}$ in terms of
the Hankel function $H_{0}^{(1)}(k_{\parallel}R_{\parallel})$ and using the
reflection formula Arfken . This enables us to extend the integration interval
to $-\infty$. The result is as follows:

$\mathcal{I}_{1}=\frac{1}{2}\oint_{C_{S}}dk_{\parallel}k_{\parallel}R_{\mathrm{TM,TM}}^{12}(k_{\parallel})H_{0}^{(1)}(k_{\parallel}R_{\parallel})e^{\mathrm{i}k_{x,1}(|x|+x_{0})},$
(160)

where $C_{S}$ is the Sommerfeld integration path depicted in Fig. 2.6.4 of
Ref. Chew . Next, we use the far-field conditions (158) and implement the
asymptotic expansion of the Hankel function Abramowitz , yielding

$\displaystyle\mathcal{I}_{1}$ $\displaystyle\sim$ $\displaystyle
e^{-\mathrm{i}\pi/4}\sqrt{\frac{1}{2\pi
R_{\parallel}}}\oint_{C_{S}}dk_{\parallel}\sqrt{k_{\parallel}}R_{\mathrm{TM,TM}}^{12}(k_{\parallel})$
(161) $\displaystyle\times
e^{\mathrm{i}k_{\parallel}R_{\parallel}}e^{\mathrm{i}k_{x,1}(x+x_{0})}\;.$

Then, by choosing the stationary phase as
$\mathrm{i}k_{\parallel}R_{\parallel}+\mathrm{i}k_{x,1}(x+x_{0})$ the saddle-
point of $\mathcal{I}_{1}$ is

$k_{\parallel s}=\frac{k_{1}R_{\parallel}}{\tilde{R}_{1}}\Rightarrow
k_{x,1\,s}=k_{1}^{2}-k_{\parallel s}^{2}\;.$ (162)

where $\tilde{R}_{1}=\sqrt{(x+x_{0})^{2}+R_{\parallel}^{2}}$. In this way,
around the saddle-point we can write

$\displaystyle\mathcal{I}_{1}$ $\displaystyle\sim$
$\displaystyle\frac{1}{2}R_{\mathrm{TM,TM}}^{12}(k_{\parallel
s})\oint_{C_{S}}dk_{\parallel}k_{\parallel}H_{0}^{(1)}(k_{\parallel}R_{\parallel})e^{\mathrm{i}k_{x,1}(x+x_{0})}\;,$
(163) $\displaystyle=$ $\displaystyle-R_{\mathrm{TM,TM}}^{12}(k_{\parallel
s})\frac{\partial}{\partial(x+x_{0})}\left(\frac{e^{\mathrm{i}k_{1}\tilde{R}_{1}}}{\tilde{R}_{1}}\right)\;.$

The latter equality is a consequence of the Sommerfeld identity Chew

$\frac{e^{\mathrm{i}k_{1}\tilde{R}_{1}}}{\tilde{R}_{1}}=\frac{\mathrm{i}}{2}\oint_{C_{S}}\frac{dk_{\parallel}k_{\parallel}}{k_{x,1}}H_{0}^{(1)}(k_{\parallel}R_{\parallel})e^{\mathrm{i}k_{x,1}(x+x_{0})}\;.$
(164)

After carrying out the partial derivative, the final form of $\mathcal{I}_{1}$
results

$\mathcal{I}_{1}\sim-\mathrm{i}R_{\mathrm{TM,TM}}^{12}(k_{1}R_{\parallel}/\tilde{R}_{1})\frac{k_{1}(x+x_{0})e^{\mathrm{i}k_{1}\tilde{R}_{1}}}{\tilde{R}_{1}^{2}}\;,$
(165)

where terms of order $\tilde{R}_{1}^{3}$ were discarded.

However, $\mathcal{I}_{1}$ (161) may have poles when the denominator of the
Fresnel coefficient $R_{\mathrm{TM,TM}}^{12}$ becomes zero. From Eq. (44) we
read that the poles are given by the roots of the following polynomial

$(\varepsilon_{2}k_{x,1}+\varepsilon_{1}k_{x,2})\mu_{1}\mu_{2}(k_{x,1}\mu_{2}+k_{x,2}\mu_{1})+\Delta_{\Theta}^{2}k_{x,1}k_{x,2}=0\;.$
(166)

By recalling the definition
$k_{x,j}=\sqrt{k^{2}_{j}-\mathbf{k}_{\parallel}^{2}}$, we note that it is a
fourth degree polynomial in $k_{\parallel}$ and is common for all involved
integrals of the electric field as Eqs. (40)-(47) show. The analysis of the
four roots of (166) shows that only the root $\varkappa$ is positive and
reproduces correctly the corresponding root of a dielectric–dielectric
interface ($\Delta_{\Theta}=0$). After computing the residue of the integrand
at $\varkappa$ via the Proposition 4.1.2 of Ref. Marsden-Tromba  and applying
the residue theorem to the integral (161), we obtain

$\displaystyle\mathcal{I}_{1,\mathrm{pole}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2\pi\mathrm{i}\varkappa}{R_{\parallel}}}\mathrm{Res}\left(R_{\mathrm{TM,TM}}^{12};\varkappa\right)$
(167) $\displaystyle\times e^{\mathrm{i}\varkappa
R_{\parallel}+\mathrm{i}\sqrt{k_{1}^{2}-\varkappa^{2}}(x+x_{0})}\;.$

Although there are explicit expressions for $\varkappa$ and the other three
roots of the polynomial (166), they are unwieldy to use. For this reason, we
restrict ourselves to study an interface without magnetic properties, i.e.
$\mu_{1}=\mu_{2}=1$. At order $\Delta_{\Theta}^{2}$ the four roots are

$\displaystyle\varkappa$ $\displaystyle\equiv$
$\displaystyle\varkappa_{1}=-\varkappa_{2}=k_{1}\sin\tilde{\theta}_{B}\;,$
(168) $\displaystyle\varkappa_{3}$ $\displaystyle=$
$\displaystyle-\varkappa_{4}=\left[\frac{\varepsilon_{1}-\varepsilon_{2}}{\sqrt{2}\Delta_{\Theta}}+\frac{3\Delta_{\Theta}(\varepsilon_{1}+\varepsilon_{2})}{4\sqrt{2}(\varepsilon_{1}-\varepsilon_{2})}\right]\frac{\omega}{c}\,,\;\;$
(169)

where

$\sin\tilde{\theta}_{B}=\sqrt{\frac{\varepsilon_{2}}{\varepsilon_{1}+\varepsilon_{2}}}\left[1+\frac{\Delta_{\Theta}^{2}\varepsilon_{1}\varepsilon_{2}}{(\varepsilon_{1}-\varepsilon_{2})^{2}(\varepsilon_{1}+\varepsilon_{2})}\right].$
(170)

Only $\varkappa$ is physically admisible for the same reasons mentioned above.
The next step is to determine the residue of the integrand at $\varkappa$. By
applying the Proposition 4.1.2 of Ref. Marsden-Tromba  and the residue
theorem, we find

$\displaystyle\mathcal{I}_{1,\mathrm{pole}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2\pi\mathrm{i}\omega}{R_{\parallel}c}}e^{\mathrm{i}\varkappa
R_{\parallel}+\mathrm{i}\sqrt{k_{1}^{2}-\varkappa^{2}}(x+x_{0})}\frac{\mathcal{N}(R_{\mathrm{TM,TM}}^{12})}{\mathcal{D}^{\prime}(R_{\mathrm{TM,TM}}^{12})}\Bigg{|}_{\varkappa}\,,$
$\displaystyle\simeq$
$\displaystyle\sqrt{\frac{2\pi\mathrm{i}\omega}{R_{\parallel}c}}e^{\mathrm{i}\varkappa
R_{\parallel}+\mathrm{i}\sqrt{k_{1}^{2}-\varkappa^{2}}(x+x_{0})}\mathcal{F}_{1}(\varepsilon_{1},\varepsilon_{2},\Delta_{\Theta})\;,$

where the prime means derivative with respect to $k_{\parallel}$,
$\mathcal{N}$ and $\mathcal{D}$ stand for the numerator and denominator of
$R_{\mathrm{TM,TM}}^{12}$ respectively. The function $\mathcal{F}_{1}$ is
explicitly written in Appendix C. In the last line only terms at order
$\Delta_{\Theta}^{2}$ were considered. After adding the results (165) and
(167) or for a non magnetic interface (165) and (B) the calculation of
$\mathcal{I}_{1}$ is completed.

By adapting this procedure to the remaining integral $\mathcal{I}_{2}$ given
by Eq. (157), we obtain

$\mathcal{I}_{2}\sim
R_{\mathrm{TM,TE}}^{12}(k_{1}R_{\parallel}/\tilde{R}_{1})\frac{e^{\mathrm{i}k_{1}\tilde{R}_{1}}}{\mathrm{i}\tilde{R}_{1}}\;,$
(172)
$\mathcal{I}_{2,\mathrm{pole}}=\sqrt{\frac{2\pi\mathrm{i}\varkappa}{R_{\parallel}}}e^{\mathrm{i}\varkappa
R_{\parallel}+\mathrm{i}\sqrt{k_{1}^{2}-\varkappa^{2}}(x+x_{0})}\mathrm{Res}\left(R_{\mathrm{TE,TM}}^{12};\varkappa\right)\,,$
(173)
$\mathcal{I}_{2,\mathrm{pole}}=\sqrt{\frac{2\pi\mathrm{i}\omega}{R_{\parallel}c}}e^{\mathrm{i}\varkappa
R_{\parallel}+\mathrm{i}\sqrt{k_{1}^{2}-\varkappa^{2}}(x+x_{0})}\frac{\mathcal{F}_{2}(\varepsilon_{1},\varepsilon_{2},\Delta_{\Theta})}{\sqrt{k_{1}^{2}-\varkappa^{2}}}\;,$
(174)

where $\mathcal{F}_{2}$ is explicitly written in Appendix C. Again the sum of
Eq. (172) and (173) or for a non magnetic interface (172) and (174) finishes
the calculation of $\mathcal{I}_{2}$.

Finally, by substituting $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ into Eq.
(20), utilizing the far-field approximation (158) by writing
$\tilde{R}_{1}\simeq R_{1}-zz^{\prime}/R_{1}$ with
$R_{1}=\sqrt{(x+x_{0})^{2}+\rho^{2}}$ and carrying out the integral over
$z^{\prime}$, one arrives at the $x$ component of Eq. (67). For the case of a
non magnetic interface, the $x$ component of the surface-wave contribution is
given by Eq. (92).

To derive the transmissive components of the electric field as described in
Eq. (87), we follow a similar procedure. However, the corresponding stationary
phase yields a fourth-degree polynomial, resulting in solutions that are
cumbersome and analytically challenging. Given the significant suppression of
transmitted radiation, in this study, we focus on distances where $x_{0}\ll
x$, enabling us to derive analytical expressions.

##  Appendix C Functions
$\mathcal{F}_{j}(\varepsilon_{1},\varepsilon_{2},\Delta_{\Theta})$ with
$j=1,2$ for the surface waves

In this appendix, we write the explicit form of the functions
$\mathcal{F}_{j}(\varepsilon_{1},\varepsilon_{2},\Delta_{\Theta})$ with
$j=1,2$ of the reflected electric field (92) and the transmitted one (97),
when non-magnetic media $\mu_{1}=\mu_{2}=1$ are considered.

$\mathcal{F}_{1}(\varepsilon_{1},\varepsilon_{2},\Delta_{\Theta})=\frac{\Delta_{\Theta}^{2}\alpha_{1}^{3/2}\alpha_{2}}{\alpha_{3}}\;,$
(175)
$\mathcal{F}_{2}(\varepsilon_{1},\varepsilon_{2},\Delta_{\Theta})=\frac{\alpha_{1}^{3/2}\alpha_{4}\Delta}{\alpha_{3}\sqrt{\varepsilon_{1}-\alpha_{1}^{2}}}\;,$
(176)

where we introduced

$\displaystyle\alpha_{1}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\varepsilon_{1}\varepsilon_{2}}{\varepsilon_{1}+\varepsilon_{2}}}\;,$
(177) $\displaystyle\alpha_{2}$ $\displaystyle=$
$\displaystyle\frac{2\alpha_{1}^{4}}{\varepsilon_{2}(\varepsilon_{1}-\varepsilon_{2})}\left[\varepsilon_{2}+\alpha_{1}^{2}\left(\varepsilon_{1}+1\right)\right]\;,$
(178) $\displaystyle\alpha_{3}$ $\displaystyle=$
$\displaystyle-\frac{(\varepsilon_{1}+\varepsilon_{2})^{2}}{\alpha_{1}}\;,$
(179) $\displaystyle\alpha_{4}$ $\displaystyle=$
$\displaystyle-2\sqrt{\varepsilon_{1}}\alpha_{1}^{2}\;.$ (180)

##  Appendix D Solving the radiated energy integrals

In this appendix, we solve the required integrals to obtain the radiated
energy of Eq. (115). Let us first calculate $d\mathcal{E}_{1}^{1}/d\omega$
being the radiated energy of the free term because it is the easiest one to
integrate. This will serve to illustrate the procedure based on Refs. OJF-LFU-
ORT ; Panofsky , which will be applied repeatedly in this appendix.
Integrating the expression (103) with respect to the corresponding solid angle
we obtain

$\displaystyle\frac{d\mathcal{E}_{1}^{(1)}}{d\omega}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\varepsilon_{0}\varepsilon_{1}}{\mu_{0}\mu_{1}}}\frac{q^{2}\omega^{2}\mu^{2}_{1}}{16\pi^{2}\varepsilon_{0}^{2}c^{4}}\left(1-\frac{1}{\beta^{2}n_{1}^{2}}\right)$
(181)
$\displaystyle\times\int_{0}^{1}d(\cos\theta)\mathcal{K}_{1}^{2}(\theta,\omega).$

The delta-like behavior of the integrand in the limit $\zeta\gg\omega/v$ shows
that the radiation is sharply localized in a main lobe around the VC angle
$\theta_{1}^{C}$ given by Eq. (90), producing a half cone in the forward
direction as Figs. 2 and 3 show. Consequently, after employing the change of
variable $u=\omega\zeta(1-vn_{1}\cos\theta)/v$, we are allowed to extend the
integration limits of the integral to $\pm\infty$ as long as we include the
maximum of the lobe situated at $u=0$. In this way, it results that

$\displaystyle\frac{d\mathcal{E}_{1}^{(1)}}{d\omega}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\varepsilon_{0}\varepsilon_{1}}{\mu_{0}\mu_{1}}}\frac{q^{2}\omega\mu^{2}_{1}\zeta}{4\pi^{2}\varepsilon_{0}^{2}c^{3}n_{1}}\left(1-\frac{1}{\beta^{2}n_{1}^{2}}\right)\int_{-\infty}^{\infty}du\frac{\sin^{2}u}{u^{2}}\;,$
(182) $\displaystyle=$
$\displaystyle\frac{q^{2}\omega\mu_{1}L}{8\pi\varepsilon_{0}c^{2}}\left(1-\frac{1}{\beta^{2}n_{1}^{2}}\right)\;,$

where we introduced the total length $L=2\zeta$ traveled by the particle.

The second term comes from the angular integration of Eq. (104),

$\displaystyle\frac{d\mathcal{E}_{1}^{(2)}}{d\omega}=-2\sqrt{\frac{\varepsilon_{0}\varepsilon_{1}}{\mu_{0}\mu_{1}}}\frac{q^{2}\omega^{2}\mu^{2}_{1}}{16\pi^{3}\varepsilon_{0}^{2}c^{4}}\left(1-\frac{1}{\beta^{2}n_{1}^{2}}\right)$
$\displaystyle\times\int
d\Omega_{1}\mathcal{K}_{1}^{2}(\theta,\omega)\mathrm{Re}\left[R_{\mathrm{TM,TM}}^{12}(\theta_{1}^{C},\phi_{1},\Delta_{\Theta})\right]\cos^{2}\phi_{1}$
$\displaystyle\times\cos\left(2k_{1}x_{0}\sin\theta_{1}^{C}\cos\phi_{1}\right).$
(183)

All the $\theta$-dependency lies only on the $\mathcal{K}_{1}$ function
enabling us to repeat the above procedure. So upon carrying out the integral
over the azimuthal angle $\phi_{1}$, we find

$\displaystyle\frac{d\mathcal{E}_{1}^{(2)}}{d\omega}$ $\displaystyle=$
$\displaystyle-\frac{2}{\pi}\frac{d\mathcal{E}_{1}^{(1)}}{d\omega}\int_{I_{\mathrm{UH}}}d\phi_{1}\cos\left(2k_{1}x_{0}\sin\theta_{1}^{C}\cos\phi_{1}\right)$
(184)
$\displaystyle\times\mathrm{Re}\left[R_{\mathrm{TM,TM}}^{12}(\theta_{1}^{C},\phi_{1},\Delta_{\Theta})\right]\cos^{2}\phi_{1}\;.$

The calculation of $d\mathcal{E}_{1}^{(3)}/d\omega$ starts from Eq. (105),
which after applying the same procedure can be written in the following
fashion

$\displaystyle\frac{d\mathcal{E}_{1}^{(3)}}{d\omega}$
$\displaystyle=\frac{2}{\pi\beta
n_{1}}\frac{d\mathcal{E}_{1}^{(1)}}{d\omega}\int_{I_{\mathrm{UH}}}d\phi_{1}\cos\phi_{1}\sin\phi_{1}$
(185)
$\displaystyle\times\mathrm{Re}\left[R_{\mathrm{TM,TM}}^{12}(\theta_{1}^{C},\phi_{1},\Delta_{\Theta})R_{\mathrm{TE,TM}}^{12\,*}(\theta_{1}^{C},\phi_{1},\Delta_{\Theta})\right],$
$\displaystyle=0\;.$

This integral vanishes by noting that the integrand is an odd function and
after a change of variable one can transform the interval of integration
$I_{\mathrm{UH}}$ to $[-\pi/2,\pi/2]$.

Finally, along the same lines and beginning with Eqs. (106) and (107), one
finds that the final expressions for $d\mathcal{E}_{1}^{(4)}/d\omega$ and
$d\mathcal{E}_{1}^{(5)}/d\omega$ are just Eqs. (119) and (120).

## References

  * (1) L. Fu, C. L. Kane and E. J. Mele, Phys. Rev. Lett. 98, 106803 (2007); L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007).
  * (2) J. E. Moore and L. Balents, Phys. Rev. B 75, 121306(R) (2007); R. Roy, Phys. Rev. B 79, 195322 (2009).
  * (3) D. Hsieh et al., Nature (London) 452, 970 (2008).
  * (4) Y. Xia et al., Nat. Phys. 5, 398 (2009); H. Zhang et al., Nat. Phys. 5, 438 (2009); Y. Ando, J. Phys. Soc. Japan 82, 102001 (2013).
  * (5) Anirban, A. Nat. Rev. Phys. 5, 267 (2023).
  * (6) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
  * (7) X. L. Qi and S. C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
  * (8) Markus Morgenstern et al., Phys. Status Solidi B 258, 2000060 (2021). DOI: 10.1002/pssb.202000060
  * (9) C. Z. Chang et al., Science 340, 167 (2013); Xufeng Kou Kou et al., Phys. Rev. Lett. 113, 137201 (2014); J. G. Checkelsky et al., Nat. Phys. 10, 731 (2014); C. Z. Chang et al., Nat. Mater. 14, 473 (2015).
  * (10) M. Mogi et al., Nat. Mater. 16, 516-521 (2017).
  * (11) L. Wu et al., Science 354, 1124-1127 (2016).
  * (12) V. Dziom et al., Nat. Commun. 8, 15197 (2017).
  * (13) X. L. Qi, T. L. Hughes, S. C. Zhang, Phys. Rev. B 78, 195424 (2008).
  * (14) K. N. Okada et al. Nat. Commun. 7, 12245 (2016).
  * (15) X. L. Qi, R. D. Li, J. D. Zang, and S. C. Zhang, Science 323, 1184 (2009).
  * (16) O. J. Franca, L. F. Urrutia and O. Rodríguez-Tzompantzi. Phys. Rev. D 99, 116020 (2019).
  * (17) O. J. Franca and Stefan Yoshi Buhmann, Phys. Rev. B 105, 155120 (2022).
  * (18) O. J. Franca and L. F. Urrutia. Revista Mexicana de Física 68, 060701 (2022).
  * (19) V. P. Jelley, Br. J. Appl. Phys. 6, 227 (1955).
  * (20) V. P. Jelley 1958 Čerenkov Radiation and its Applications (Oxford: Pergamon).
  * (21) P. A. Čerenkov, Dokl. Akad. Nauk SSSR 2, 451 (1934).
  * (22) S. I. Vavilov, Dokl. Akad. Nauk SSSR 2, 457 (1934).
  * (23) I. M. Frank and I. E. Tamm, Dokl. Akad. Nauk. 14, 107 (1937) [Compt. Rend. (Dokl) 14, 109 (1937)].
  * (24) Ypsilantis, T. and Seguinot, J. Theory of ring imaging Cherenkov counters. Nucl. Instrum. Meth. A 343, 30–51 (1994).
  * (25) See for example the section 35.5 in C. Patrignani et al. (Particle Data Group), Chin. Phys. C 40, 100001 (2016) and 2017 update.
  * (26) Chamberlain, O., Segre, E., Wiegand, C. and Ypsilantis, T. Observation of antiprotons. Phys. Rev. 100, 947 (1955).
  * (27) Aubert, J. J. et al. Experimental observation of a heavy particle. J. Phys. Rev. Lett. 33, 1404 (1974).
  * (28) J. Tao, Q. J. Wang, J. Zhang and Y. Luo. Scientific Reports 6,30704 (2016).
  * (29) T. H. O’Dell, The Electrodynamics of Magneto-Electric Media (North-Holland, Amsterdam, 1970); L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Continuous Media (Course of Theoretical Physics Vol. 8) (Oxford: Pergamon Press, 1984).
  * (30) A. A. Kolomenskii, Zh. Eksp. Teor. Fiz. 24, 167 (1953).
  * (31) Sergey N. Galyamin, Anton A. Peshkov and Andrey V. Tyukhtin, Phys. Rev. E 88, 013206 (2013).
  * (32) A. E. Spinelli, M. Ferdeghini, C. Cavedon, E. Zivelonghi, R. Calandrino, A. Fenzi, A. Sbarbati, and F. Boschi, J. Biomed. Opt. 18, 020502 (2013).
  * (33) H. Hu, X. Lin, L. J. Wong et al, eLight 2, 2 (2022).
  * (34) V. G. Veselago, Soviet Physics Uspekhi 10, 509 (1968).
  * (35) B. M. Bolotovskii, Sov. Phys. Usp. 4, 781 (1962).
  * (36) M. -C. Chang and M. -F. Yang, Phys. Rev. B 80, 113304 (2009).
  * (37) Y. N. Obukhov and F. W. Hehl, Phys. Lett. A 341, 357 (2005).
  * (38) Stefan Scheel and Stefan Yoshi Buhmann, Acta Phys. Slovaca 58, 675 (2008).
  * (39) J. A. Crosse, Sebastian Fuchs and Stefan Yoshi Buhmann, Phys. Rev. A 92, 063831 (2015).
  * (40) A. M. Essin, J. E. Moore, and D. Vanderbilt, Phys. Rev. Lett. 102, 146805 (2009).
  * (41) F. Wilczek, Phys. Rev. Lett. 58, 1799 (1987).
  * (42) Qing-Dong Jiang and Frank Wilczek, Phys. Rev. B 99, 165402 (2019).
  * (43) L. D. Barron, Chemical Physics Letters 123, 5 (1986).
  * (44) L. D. Barron, J. Am. Chem. Soc. 108, 18 (1986).
  * (45) A. Martín-Ruiz, M. Cambiaso and L. F. Urrutia, Phys. Rev. D 92, 125015 (2015).
  * (46) A. Martín-Ruiz, M. Cambiaso and L. F. Urrutia. EPL 113, 60005 (2016).
  * (47) T. H. O’Dell, The Electrodynamics of Magneto-Electric Media (North-Holland, Amsterdam, 1970).
  * (48) L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii. Electrodynamics of Continuous Media, Course of Theoretical Physics Vol. 8 (Pergamon Press, Oxford, 1984).
  * (49) F.W. Hehl, Y.N. Obukhov, J.-P. Rivera and H. Schmid, Phys. Lett. A 372, 1141-1146 (2008).
  * (50) R. M. Khan, Br. J. Appl. Phys. 18, 1443 (1967).
  * (51) J. A. Crosse, Sebastian Fuchs and Stefan Yoshi Buhmann, Phys. Rev. A 92, 063831 (2015).
  * (52) Sebastian Fuchs, J. A. Crosse and Stefan Yoshi Buhmann, Phys. Rev. A 95, 023805 (2017).
  * (53) Stefan Yoshi Buhmann, Dispersion Forces I. Macroscopic Quantum Electrodynamics and Ground State Casimir, Casimir-Polder and van der Waals Forces (Springer, Berlin Heidelberg, 2012).
  * (54) Weng Cho Chew, Waves and Fields in Inhomogenous Media (IEEE Press Series on Electromagnetic Wave Theory. New York, 1990).
  * (55) L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
  * (56) W. C. Chew, IEEE Trans. Antennas Propag. Vol. 36, No. 11, 1654-1657 (1988).
  * (57) J. R. Wait, Electromagnetic waves in stratified media, Revised edition including supplemented material (Pergamon Press, Oxford, 1970).
  * (58) L. Novotny and B. Hecht, Principles of Nano Optics, (Cambridge University Press, 2006).
  * (59) L. Chen and S. Wan, Phys. Rev. B 85, 115102 (2012); Wenjie Nie, Ran Zeng, Yueheng Lan, and Shiyao Zhu, Phys. Rev. B 88, 085421 (2013); R. Zeng et al., Phys. Lett. A 380, 2861 (2016).
  * (60) J. D. Jackson. Classical Electrodynamics (Wiley, 3rd edition, New York, 1999).
  * (61) J. Schwinger, L. L. DeRaad, Jr., K. A. Milton, Wu-yang Tsai, Classical Electrodynamics (Westview Press, Boulder, Colorado, 1998).
  * (62) M. J. Lighthill. _Introduction to Fourier Analysis and Generalised Functions._ Cambridge University Press, London, 1958.
  * (63) S. S. Batsanov, E. D. Ruchkin, I. A. Poroshina, Refractive Indices of Solids, (SpringerBriefs in Applied Sciences and Technology, Singapore, 2016.
  * (64) J. -P. Rivera, Eur. Phys. J. B 71, 299 (2009).
  * (65) W. K. H. Panofsky and M. Phillips. _Classical Electricity and Magnetism._ Addison-Wesley Publishing, Inc., 2nd ed., 1962, USA.
  * (66) Stefan Scheel and Stefan Yoshi Buhmann, Phys. Rev. A 80, 042902 (2009).
  * (67) Gregor Pieplow and Carsten Henkel, J. Phys.: Condens. Matter 27, 214001 (2015).
  * (68) Di Xiao et al., Phys. Rev. Lett. 120, 056801 (2018); M. Mogi et al., Sci. Adv. 3, 1669 (2017); Nicodemos Varnava and David Vanderbilt, Phys. Rev. B 98, 245117 (2018).
  * (69) P. Sikivie, Rev. Mod. Phys. 93, 15004 (2021).
  * (70) Dennis M. Nenno, Christina A. C. Garcia, Johannes Gooth, Claudia Felser and Prineha Narang, Nat. Rev. Phys. 2, 682 (2020).
  * (71) David J.E. Marsh, Kin Chung Fong, Erik W. Lentz, Libor Šmejkal and Mazhar N. Ali, Phys. Rev. Lett. 123, 121601 (2019).
  * (72) Jan Schütte-Engel, David J.E. Marsh, Alexander J. Millar, Akihiko Sekine, Francesca Chadha-Day, Sebastian Hoof, Mazhar N. Ali, Kin Chung Fong, Edward Hardy and Libor Šmejkal, JCAP 08, 066 (2021).
  * (73) Henrik S. Røising, Benjo Fraser, Sinéad M. Griffin, Sumanta Bandyopadhyay, Aditi Mahabir, Sang-Wook Cheong and Alexander V. Balatsky, Phys. Rev. Research 3, 033236 (2021).
  * (74) G. Arfken and H. Weber. Mathematical Methods for Physicists. Elsevier Academic Press. 6a. ed. EUA, 2005.
  * (75) M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover Publications, New York, 1972.
  * (76) J. E. Marsden and A. Tromba. Vector Calculus. Pearson Addison-Wesley. 5a. ed. EUA, 2004.