#  Čerenkov radiation in vacuum from a superluminal grating

Daigo Oue  The Blackett Laboratory, Department of Physics, Imperial College
London, Prince Consort Road, Kensington, London SW7 2AZ, United Kingdom  Kun
Ding  The Blackett Laboratory, Department of Physics, Imperial College London,
Prince Consort Road, Kensington, London SW7 2AZ, United Kingdom  Department of
Physics, State Key Laboratory of Surface Physics, and Key Laboratory of Micro
and Nano Photonic Structures (Ministry of Education), Fudan University,
Shanghai 200438, China  J. B. Pendry  The Blackett Laboratory, Department of
Physics, Imperial College London, Prince Consort Road, Kensington, London SW7
2AZ, United Kingdom

###### Abstract

Nothing can physically travel faster than light in vacuum. This is why it has
been considered that there is no Čerenkov radiation (ČR) without an effective
refractive index due to some background field. In this Letter, we
theoretically predict ČR in vacuum from a spatiotemporally modulated boundary.
We consider the modulation of traveling wave type and apply a uniform
electrostatic field on the boundary to generate electric dipoles. Since the
induced dipoles stick to the interface, they travel at the modulation speed.
When the grating travels faster than light, it emits ČR. In order to
quantitatively examine this argument, we need to calculate the field scattered
at the boundary. We utilise a dynamical differential method, which we
developed in the previous paper, to quantitatively evaluate the field
distribution in such a situation. We can confirm that all scattered fields are
evanescent if the modulation speed is slower than light while some become
propagating if the modulation is faster than light.

Introduction.—Čerenkov radiation (ČR) is radiation from a charged particle
moving faster than light in a medium, which was originally observed by
Čerenkov in 1934 [1] and then theoretically studied by Tamm and Frank [2]. It
has been observed in various systems, including metamaterials and photonic
crystals where the speed of light is effectively suppressed and hence the
threshold for ČR [3, 4, 5, 6]. There is also ČR into surface modes such as
surface plasmon polaritons and Dyankov waves [7, 8, 9]

Even uncharged moving particles can emit ČR if electrically or magnetically
polarised, which is closely related to the friction induced by electromagnetic
fields [10, 11, 12]. Many attempts to generate ČR in linear optics were
focused on slowing down the speed of light by engineering the medium’s
dispersion relation as in the case of photonic crystals and metamaterials. On
the other hand, it has been reported that ČR is emitted not only from
physically moving dipoles but also from ones induced, for example, by moving
light foci [13, 14, 15] and solitons in nonlinear optical fibres [16, 17, 18]
or micro resonators [19, 20, 21].

ČR is composed of coherent multi-frequency components propagating in the same
direction. It is for this property that ČR has not only attracted scientific
interest but also been applied to other research fields. ČR emitted by cosmic
rays has been captured in large facilities such as super-Kamiokande and played
vital roles in astrophysics and high energy physics [22]. As mentioned above,
ČR can be generated in nonlinear media. It is utilised as an optical frequency
comb, as a tuneable and broadband source and is expected to be utilised in
spectroscopy or metrology [23, 24, 19].

There are many reports of ČR in media as reviewed above; however, there is a
limited number of studies on ČR in vacuum. This is because nothing can
physically move faster than light in vacuum. Previous studies show that
introducing some background such as external electromagnetic or Chern-Simons
fields induces effective refractive index to lower the ČR threshold in vacuum
below the light barrier [25, 26, 27, 28].

In this Letter, we propose a mechanism for ČR in vacuum without introducing
any effective index. We consider a single interface system composed of vacuum
and a dielectric (Figure 1).

Figure 1:  Single interface system composed of a dielectric and vacuum. The
interface is under the spatiotemporal modulation of traveling wave type (1).
The orthonormal tangential vectors, $\vec{t}_{1,2}$ can be calculated from the
interface profile. The permittivities above and below the interface are
denoted as $\epsilon^{\scriptscriptstyle{>}}$ and
$\epsilon^{\scriptscriptstyle{<}}$, respectively. We apply a uniform
electrostatic field $\vec{\mathcal{E}}_{\mathbf{x},z}^{\mathrm{in}}$ on the
modulated interface so that there are induced dipoles which travel on the
interface due to its profile of the traveling wave type. The velocity of the
grating $v_{\mathrm{ph}}=\Omega/g$ can be tuned by two independent parameters
so that it can exceed the speed of light. When the speed is faster than light,
the induced dipoles emit Čerenkov radiation.

By spatiotemporally modulating its interface, we can generate an interface
profile of a travelling wave type,

$\displaystyle
a_{\mathbf{x}}=A\sin(\mathbf{q}\cdot\mathbf{x})=A\sin\left[g\left(x-\frac{\Omega}{g}t\right)\right],$
(1)

where we have defined a three-component vector $\mathbf{x}\equiv\\{x,y,ict\\}$
and a reciprocal vector $\mathbf{q}\equiv\\{g,0,i\Omega/c\\}$ for the sake of
convenience, and $v_{\mathrm{ph}}\equiv\Omega/g$ is the sliding speed of the
profile. Note that $\Omega$ and $g$ are independent modulation parameters
(temporal and spatial modulation frequencies), and thus the sliding speed
$v_{\mathrm{ph}}$ is not limited by the speed of light. Since we are focusing
on dielectrics, the permeability is assumed to be unity everywhere ($\mu=1$).
The permittivity is given by means of the interface profile,

$\displaystyle\epsilon_{\mathbf{x},z}=\alpha\Theta(a_{\mathbf{x}}-z)+\epsilon^{\scriptscriptstyle{>}},$
(2)

where we denote the permittivities of the upper and lower media as
$\epsilon^{\raisebox{1.04996pt}[0.0pt][0.0pt]{$\scriptscriptstyle{>}$}\raisebox{-1.04996pt}[0.0pt][0.0pt]{$\scriptscriptstyle{<}$}}$,
the permittivity difference as
$\alpha\equiv\epsilon^{\scriptscriptstyle{<}}-\epsilon^{\scriptscriptstyle{>}}$
and the Heaviside unit step function as $\Theta(z)$.

When an electrostatic field
$\vec{\mathcal{E}}^{\mathrm{in}}=E_{\mathrm{in}}\vec{u}_{y}$ is applied to
this configuration, electric dipoles are induced on the interface. Following
the traveling type profile of the interface, the induced dipoles move
virtually along the interface. If the profile velocity and hence that of the
induced dipoles is faster than light, the induced dipoles may emit ČR.

Our system is periodic in the $x$ direction and in time, and thus the
radiation wavenumber in the $x$ direction and the frequency in our system are
written $k_{x,m}=k_{x}+mg$ and $\omega_{m}=\omega+m\Omega$ in accordance with
the Floquet-Bloch theory, where $m=0,\pm 1,\pm 2,\cdots$ is the diffraction
order. Note that here we have multi-frequency radiation in contrast with
ordinary diffraction. The propagarion direction of the $m$th order diffraction
is $\theta_{m}^{\tau}=\cos^{-1}(ck_{x,m}/\omega_{m}\sqrt{\epsilon^{\tau}})$ in
each medium. Since we have a uniform electrostatic field ($k_{x}=0$,
$\omega=0$), all the polychromatic waves in the different diffraction orders
propagate in the same direction in each medium,
$\theta_{m}^{\tau}\rightarrow\cos^{-1}(c/v_{\mathrm{ph}}\sqrt{\epsilon^{\tau}})\equiv\theta_{\scriptscriptstyle{\mathrm{\check{C}R}}}^{\tau}$.
We quantitatively analyse such a situation below.

Modulation-induced source at the interface.—Here, we briefly review a
dynamical differential method, which we developed in the previous work [29],
and calculate the diffraction at the interface subject to modulations in space
and time. Our previous work is based on the coordinate translation method
originally proposed by Chandezon et al. [30, 31], where they consider static
corrugated interfaces and match Maxwell’s boundary conditions directly at the
interfaces with the help of differential geometry. Using this method, it is
straightforward to take the structure of the interfaces into consideration,
and it has been utilised to calculate structured surfaces of various media,
including anisotropic, plasmonic and dielectric materials [32, 33, 34, 35, 36]
It is also worth noting that there is a series of studies, which confirm that
the method works well for smooth shallow corrugations and propose possible
ways to improve the method so that they can handle deep corrugation even with
sharp edges [37, 38, 39, 40, 41, 42, 43]. In these works, local distortion of
the coordinate systems is applied instead of the global translation of the
coordinate in order to improve the convergence. Since we are interested in the
time-dependent corrugation of the traveling wave type, we assume that the
corrugation depth is small and safely work on the simple global coordinate
translation.

In order to calculate diffraction at the structured interface, we need
relevant boundary conditions. By integrating Maxwell-Heaviside equations over
two kinds of path, $\Gamma_{1,2}$, which enclose the interface as shown in
Figure 2, we can derive the boundary conditions,

$\displaystyle\begin{cases}{}\eta\vec{t}_{1}\cdot\left(\vec{\mathcal{H}}_{\mathbf{x},a_{\mathbf{x}}+0}-\vec{\mathcal{H}}_{\mathbf{x},a_{\mathbf{x}}-0}\right)&=\vec{t}_{2}\cdot\vec{j}_{\mathbf{x}}^{\mathrm{sou}}\\\
\vec{t}_{2}\cdot\left(\vec{\mathcal{E}}_{\mathbf{x},a_{\mathbf{x}}+0}-\vec{\mathcal{E}}_{\mathbf{x},a_{\mathbf{x}}-0}\right)&=0.\end{cases}$
(3)

Here, we have a modulation-induced source term,

$\displaystyle\vec{j}_{\mathbf{x}}^{\mathrm{sou}}$
$\displaystyle=\cfrac{\dot{a}_{\mathbf{x}}}{c}\alpha\vec{\mathcal{E}}_{\mathbf{x},a_{\mathbf{x}}-0},$
(4)

which is finite if there is time dependence (i.e. $\dot{a}_{\mathbf{x}}\neq
0$). This source term is responsible for radiation from the interface. Note
that we take an unit system such that
$Z_{0}\equiv\sqrt{\epsilon_{0}}{\mu_{0}}=1$ for simplicity. The orthonormal
tangential vectors of the interface, $\vec{t}_{1,2}$, can be given by means of
the interface profile,

$\displaystyle\vec{t}_{1}=\cfrac{\vec{u}_{x}+a_{\mathbf{x}}^{\prime}\vec{u}_{z}}{\sqrt{1+{a_{\mathbf{x}}^{\prime}}^{2}}},\quad\vec{t}_{2}=\vec{u}_{y},$
(5)

and the electric and magnetic fields in real space as
$\vec{\mathcal{E}}_{\mathbf{x},z}$ and $\vec{\mathcal{H}}_{\mathbf{x},z}$.
Note that we have a factor of $\eta=\sqrt{1+{a_{\mathbf{x}}^{\prime}}^{2}}$ in
front of $\vec{t}_{1}$ in Eq. (3), which corresponds to the spatial variation
(expansion or contraction) of the differential line element due to the
coordinate translation, while we do not have that factor at $\vec{t}_{2}$
because the translation does not affect the line element in the direction.

Figure 2:  Integration paths leading to the boundary conditions at the
modulated interface. The modulation induced source,
$\vec{j}_{\mathbf{x}}^{\mathrm{sou}}$, comes into contribution.

Since our system is periodic in space and time, we can substitute Floquet-
Bloch type solutions in the upper and lower media, e.g.
$\vec{\mathcal{E}}_{\mathbf{x},z}=\sum_{m}E_{\mathbf{k}_{m},z}\vec{t}_{2}e^{i\mathbf{k}_{n}\cdot\mathbf{x}}\
(\mathbf{k}_{m}=\mathbf{k}+m\mathbf{q},\
\mathbf{k}\equiv\\{k_{x},0,ik_{0}=i\omega/c\\})$, to obtain simultaneous
equations in the reciprocal space,

$\displaystyle\begin{pmatrix}\mathsf{N}_{\mathbf{k}}^{{\scriptscriptstyle{+}}{\scriptscriptstyle{>}}}&-(\mathsf{N}_{\mathbf{k}}^{{\scriptscriptstyle{-}}{\scriptscriptstyle{<}}}+\mathsf{L}_{\mathbf{k}})\\\
\mathsf{M}_{\mathbf{k}}^{{\scriptscriptstyle{+}}{\scriptscriptstyle{>}}}&-\mathsf{M}_{\mathbf{k}}^{{\scriptscriptstyle{-}}{\scriptscriptstyle{<}}}\end{pmatrix}\begin{pmatrix}\mathbb{E}_{\mathbf{k}}^{{\scriptscriptstyle{-}}{\scriptscriptstyle{<}}}\\\
\mathbb{E}_{\mathbf{k}}^{{\scriptscriptstyle{+}}{\scriptscriptstyle{>}}}\end{pmatrix}=\begin{pmatrix}-\mathsf{N}_{\mathbf{k}}^{{\scriptscriptstyle{-}}{\scriptscriptstyle{>}}}\mathbb{E}_{\mathbf{k}}^{{\scriptscriptstyle{-}}{\scriptscriptstyle{>}}}\\\
-\mathsf{M}_{\mathbf{k}}^{{\scriptscriptstyle{-}}{\scriptscriptstyle{>}}}\mathbb{E}_{\mathbf{k}}^{{\scriptscriptstyle{-}}{\scriptscriptstyle{>}}}\end{pmatrix}$
(6)

where we collect the modal amplitude in each diffraction order to form
${\mathbb{E}_{\mathbf{k}}^{\sigma\tau}}=\begin{pmatrix}\cdots&E_{\mathbf{k}_{-1}}^{\sigma\tau}&E_{\mathbf{k}_{0}}^{\sigma\tau}&E_{\mathbf{k}_{+1}}^{\sigma\tau}&\cdots\end{pmatrix}^{\intercal}$.
We have specified the upper and lower media by
$\tau={\raisebox{1.49994pt}[0.0pt][0.0pt]{$\scriptscriptstyle{>}$}\raisebox{-1.49994pt}[0.0pt][0.0pt]{$\scriptscriptstyle{<}$}}$,
and the propagating direction is labeled by $\sigma=\pm$. Thus, we can regard
$E_{\mathbf{k}_{m}}^{{\scriptscriptstyle{-}}{\scriptscriptstyle{>}}}$ as the
incident components from the upper side while
$E_{\mathbf{k}_{m}}^{{\scriptscriptstyle{+}}{\scriptscriptstyle{>}}}$ and
$E_{\mathbf{k}_{m}}^{{\scriptscriptstyle{-}}{\scriptscriptstyle{<}}}$ as the
diffracted components in the upper and lower sides. The effects of the source
term and interface geometry are encoded in the $\mathsf{L}$, $\mathsf{M}$ and
$\mathsf{N}$ coefficients matrices, whose elements read

$\displaystyle[\mathsf{L}_{\mathbf{k}}]_{lm}$
$\displaystyle=\frac{\alpha}{c}\frac{(l-m)\Omega}{-K_{\mathbf{k}_{m}}^{{\scriptscriptstyle{<}}}}\times\operatorname{sgn}(\omega_{m})\frac{k_{x,m}}{|k_{x,m}|}J_{l-m}(\phi_{\mathbf{k}_{m}}^{{\scriptscriptstyle{-}}{\scriptscriptstyle{<}}}),$
(7) $\displaystyle[\mathsf{M}_{\mathbf{k}}^{\sigma\tau}]_{lm}$
$\displaystyle=\frac{k_{x,m}}{|k_{x,m}|}\operatorname{sgn}(\omega_{m})J_{l-m}(\phi_{\mathbf{k}_{m}}^{\sigma\tau}),$
(8) $\displaystyle[\mathsf{N}_{\mathbf{k}}^{\sigma\tau}]_{lm}$
$\displaystyle=\left(\frac{\sigma
K_{\mathbf{k}_{m}}^{\tau}}{k_{x,m}}-\frac{(l-m)g}{\sigma
K_{\mathbf{k}_{m}}^{\tau}}\right)\frac{|k_{x,m}|}{|k_{0,m}|}J_{l-m}(\phi_{\mathbf{k}_{m}}^{\sigma\tau}).$
(9)

Here, $J_{m}$ is the $m$th order Bessel function of the first kind, and we
have defined the wavenumber in the $z$ direction,

$\displaystyle
K_{\mathbf{k}}\equiv\operatorname{sgn}(\omega)\operatorname{Re}\sqrt{\epsilon^{\tau}{k_{0}}^{2}-{k_{x}}^{2}}+i\operatorname{Im}\sqrt{\epsilon^{\tau}{k_{0}}^{2}-{k_{x}}^{2}},$
(10)

and the corresponding propagating phase
$\phi_{\mathbf{k}_{m}}^{\sigma\tau}\equiv\sigma K_{\mathbf{k}_{m}}^{\tau}A$.
By truncating $\mathsf{M},\mathsf{N}$ and $\mathsf{L}$ matrices to finite rank
ones (i.e. $|l|,|m|\leq m_{c}$), we can numerically invert the matrix on the
left hand side and evaluate the diffraction amplitudes.

Čerenkov radiation.—We reconstruct the diffracted field distribution in real
space after obtaining the diffraction amplitudes. Our interest is to apply a
uniform electrostatic field ($\omega=0,k_{x}=0$). In Figure 3, we show
snapshots of the field distributions for various modulation speeds and the
corresponding cross section plots in the far field region.

Figure 3:  Snapshots (middle row) and their cross sections (upper and lower
rows) of the field distributions in the subluminal and superluminal regimes.
The intensity of the total electric field
$|\vec{\mathcal{E}}_{\mathbf{x},z}|^{2}$ is plotted. (i) Subluminal regime in
the dielectric and vacuum sides,
$v_{\mathrm{ph}}=\Omega/g=0.2c<c/\sqrt{\epsilon}$. (ii) Superluminal in the
dielectric side and subluminal in the vacuum side,
$c/\sqrt{\epsilon}<v_{\mathrm{ph}}=0.8c<c$. (iii) Superluminal regime in both
sides, $c<v_{\mathrm{ph}}=1.2c$. The modulation parameters are given as
following: the spatial frequency $g=2\pi\ [\mathrm{\mu m^{-1}}]$, the
modulation depth $2A=100\ [\mathrm{nm}]$. The input amplitude is
$E_{\mathrm{in}}=1\ \mathrm{[V\cdot m^{-1}]}$. The permittivities are
$\epsilon^{\scriptscriptstyle{>}}=1$ and
$\epsilon^{\scriptscriptstyle{<}}=\epsilon=2.25$. The cutoff is $m_{c}=10$ so
that we take $2m_{c}+1=21$ diffracted waves into account. The colorbars for
(ii) and (iii) are common and shown on the right of (iii). Note that the scale
of the colorbar of (i) is different from those of (ii) and (iii), and the
horizontal axes and the vertical axes of the color plots are normalised by the
spatial period $g$ of the modulation.

When the modulation speed is slower than that of light in the dielectric
(subluminal regime), the distribution is uniform in the far field region both
on the dielectric and vacuum sides. This is because, in the subluminal regime
($v_{\mathrm{ph}}=\Omega/g<c/\sqrt{\epsilon}$), the wavenumber in the $z$
direction is imaginary for each diffraction order,

$\displaystyle
K_{\mathbf{k}_{m}}^{\tau}=\frac{|m|g\sqrt{\epsilon^{\tau}}}{c}i\operatorname{Im}\sqrt{{v_{\mathrm{ph}}}^{2}-\left(\frac{c}{\sqrt{\epsilon^{\tau}}}\right)^{2}},$
(11)

and hence all diffracted fields exponentially decay.

On the other hand, once the modulation speed is faster than light in the
dielectric and vacuum (superluminal regime), we can find the Čerenkov type
patterns (i.e. propagating waves) in each medium. This is because, unlike in
the subluminal regime, the wavenumber in the $z$ direction is real for each
diffraction order. We can confirm that all diffraction modes in each medium
propagate in the same direction,

$\displaystyle\tan\theta_{m}^{\tau}\equiv\frac{K_{\mathbf{k}_{m}}^{\tau}}{k_{x,m}}=\frac{\sqrt{\epsilon^{\tau}}}{c}\operatorname{Re}\sqrt{{v_{\mathrm{ph}}}^{2}-\left(\frac{c}{\sqrt{\epsilon^{\tau}}}\right)^{2}},$
(12)

Note that the far right hand side does not depend on the diffraction order
$m$. Therefore, different diffraction orders with different frequencies
$\omega_{m}=m\Omega$ are superposed to form pulses. This feature is more
significant in the dielectric side (see the cross section plots in Figure 3).

What we have observed here is radiation from the virtually moving dipoles
induced by the electrostatic input on the interface. The evenly spaced, moving
dipoles collectively emit radiation in the far field, and thus the field
pattern is a plane wave. We can also mathematically understand the radiation
as emission from the source term on the right hand side of Eq. (3), which is
induced by the temporal modulation of the interface ($\dot{a}_{\mathbf{x}}\neq
0$). Since the strength of the source is proportional to the temporal
modulation frequency, the emitted field is stronger as the modulation speed
$v_{\mathrm{ph}}$ increases in Figure 3.

Conclusions.—In this Letter, we have proposed a mechanism for the Čerenkov
radiation in a system which consists of a single interface. The spatiotemporal
modulation of its interface realises the interface profile of a travelling
wave. By applying an electrostatic field, electric dipoles are induced on the
interface, and move virtually on the interface due to the travelling wave
profile. The profile slides at its phase velocity $v_{\mathrm{ph}}=\Omega/g$,
where $\Omega$ and $g$ are independent modulation parameters, temporal and
spatial modulation frequencies. Thus, the sliding speed of the profile
$v_{\mathrm{ph}}$ is not limited by the speed of light. If the profile
velocity and hence that of the induced dipoles is faster than light, they emit
ČR.

As for the experimental implementation, the key is the spatiotemporal
modulation of the interface between two media. One can use acoustic techniques
to spatiotemporally modulate the surfaces of materials. According to the
studies in acoustic communities [44, 45], surface displacement of the order of
$10\ \mathrm{nm}$ and ultrafast modulation up to $1\ \mathrm{THz}$ are
achievable through acoustic-optical modulation. Using soft materials such as
liquids is one way to generate large surface displacement up to micrometer
scale [46, 47, 48].

Another possibility is electrostatic modulation of permittivities of
atomically thin materials placed at interfaces. Those thin materials can be
regarded as infinitely thin sheets with finite conductivities [49]. Recently,
several studies have revealed that the permittivity profile of graphene can be
electrically modulated [50, 51, 52]. The modulation can be performed at high
frequency in space and time so that graphene is another candidate to realise
our proposed effects although we should take it into consideration that the
optical response is dominated by two dimensional Dirac electrons [49].

###### Acknowledgements.

D.O. is funded by the President’s PhD Scholarships at Imperial College London.
K.D. and J.B.P. acknowledges support from the Gordon and Betty Moore
Foundation.

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