# Electrodynamics of Accelerated-Modulation Space-Time Metamaterials

Amir Bahrami  Department of Electrical Engineering, KU Leuven, Leuven, Belgium
Zoé-Lise Deck-Léger  Department of Electrical Engineering, Polytechnique
Montréal, Montréal, Quebec, Canada  Christophe Caloz
<EMAIL_ADDRESS>Department of Electrical Engineering, KU Leuven,
Leuven, Belgium

###### Abstract

Space-time varying metamaterials based on uniform-velocity modulation have
spurred considerable interest over the past decade. We present here the first
extensive investigation of _accelerated-modulation_ space-time metamaterials.
Using the tools of general relativity, we establish their electrodynamic
principles and describe their fundamental phenomena, in comparison with the
physics of moving-matter media. We show that an electromagnetic beam
propagating in an accelerated-modulation metamaterial is bent in its course,
which reveals that such a medium curves space-time for light, similarly to
gravitation. Finally, we illustrate the vast potential diversity of
accelerated-modulation metamaterial by demonstrating related Schwarzschild
holes.

††preprint: APS/123-QED

##  I Introduction

Space-Time Modulation 111The noun “modulation” in the term “space-time
modulation metamaterials” should _not_ be replaced by the past participle
“modulated” because “modulated metamaterial” would erroneously suggest that
the metamaterial preexists modulation, whereas it is really _formed_ by the
modulation. metamaterials are metamaterials that are formed by varying
(modulating) some parameter (e.g., the refractive index) of a host medium in
both space and time [2, 3, 4, 5, 6, 7, 8, 9, 10]. The modulated parameter may
be of various possible natures, including electronic, optical, acoustic,
mechanical, thermal and chemical [11, 12, 13], while the modulation is
typically provided by an external drive. These metamaterials are thus _moving-
perturbation_ media, involving no net transfer of atoms and molecules, and may
hence be seen as the modulation counterparts of _moving-matter_ media, which
are simply referred to as moving media in the literature and whose basic
electrodynamics was discovered between the early 19${}^{\text{th}}$ and early
20${}^{\text{th}}$ centuries [14, 15, 16, 17]. Space-time modulation
metamaterials share most of the electrodynamics of moving media, including
Doppler shifting [18], Bradley aberrations [19] and light deflection [20],
Fresnel-Fizeau drag [14, 15] and wave-compression amplification [20, 21, 22],
as shown in a number of studies [23, 24, 25, 26, 27, 28]. However, they
encompass extra physical regimes, such as instantaneous [29, 30, 31, 32, 33]
and superluminal [34, 3, 35, 4, 36, 6, 37, 38] responses, and offer drastic
advantages in terms of potential applications, particularly the dispensability
of cumbersome moving parts and the easy access to relativistic velocities and
accelerations.

The quasi-totality of the research on space-time modulation metamaterials to
date has pertained to _uniform-velocity_ modulation. Removing the restriction
of velocity uniformity by introducing _accelerated-modulation_ would naturally
imply greater diversity and hence pave the way for novel physics and
technology. The related enhancement might perhaps be compared to that gained
by extending gravitation-less systems to gravity systems, as done from special
relativity [20, 39] to general relativity [40, 41, 42], or, according to the
principle of equivalence [43], by extending uniformly moving media to
accelerated media. The resulting metamaterials would feature some similarities
with conventional gravity analogs [44, 45] and modulated devices [11, 12], but
also transcend them via the incorporation of more sophisticated space-time
metrics.

##  II Accelerated Space-Time Structures

Figure 1 shows the space (top panels) and space-time (bottom panels)
structures of the accelerated-matter and accelerated-modulation media. It
specifically corresponds to the regime of _constant proper acceleration_ and
_uniform direction of motion_ , which will pertain to most of the paper. This
regime is chosen because it is the simplest possible acceleration regime while
sufficing to reveal the most fundamental physics of accelerated-modulation
space-time metamaterials. The last part of the paper, on analog holes, will
involve a more complex acceleration regime, featuring _varying acceleration_
and _nonuniform, spherical direction of motion_ , but featuring a structure
that is locally similar to that in Fig. 1(b) and basic principles that draw
from the related physics.

Figure 1: Accelerated- (a) matter medium and (b) modulation medium, with
constant proper acceleration (Rindler metric) and uniform direction of motion
($z$), in space at time $t=t_{0}$ (top panels) and in space-time at any
position $x$ (bottom panels). The two media are shown as bounded (on their
left side) to emphasize their evolution, in a host medium of refractive index
$n_{0}$. The acceleration, corresponding to the constant proper acceleration
($a^{\prime}$), is inversely proportional to the curvature of the curves
(Rindler hyperbolas) in the bottom panels, and is hence time-dependent,
$a=a(z,t)$, with $\mathrm{sgn}(a)=\mathrm{sgn}(a^{\prime})$ [see Appendix A].

Figure 1(a) represents the structure of an accelerated-matter medium. Such a
medium is typically an object that has been propelled by a mechanical force,
and the moving entities are the atoms and molecules [shown in the insets of
Fig. 1(a)] that form the object. The motion of these entities induces
magnetoelectric coupling [16], which generally makes the medium bianisotropic,
with well-known tensorial permittivity, permeability and coupling parameters,
$\overline{\overline{\epsilon}}$, $\overline{\overline{\mu}}$ and
$\overline{\overline{\raisebox{2.0pt}{$\chi$}}}$ [46].

Figure 1(b) depicts the structure of an accelerated-modulation medium. Such a
medium may be formed, for instance, by illuminating a dielectric/piezoelectric
slab with a circular phase-front optical/acoustic pump, or by switching a
varactor-loaded artificial transmission line structure according to a
nonuniform voltage sequence. Here, the atoms and molecules that form the
structure [shown in the insets Fig. 1(b)] do _not_ move; what moves is only
the modulation, which is a traveling-wave, typically sinusoidal, perturbation
of the refractive index. We shall assume throughout the paper that the
structure is operating in the _metamaterial regime_ , where the modulation is
subwavelength ($\lambda_{\mathrm{modulation}}\ll\lambda_{\mathrm{wave}}$) and
subperiod ($T_{\mathrm{modulation}}\ll T_{\mathrm{wave}}$) [9, 24, 25], so
that the striped structure in Fig. 1(b) can be reduced, by proper averaging,
to a _homogeneous medium_ 222The homogeneous metamaterial regime is the long-
wavelength and long-period (viewpoint of the wave) regime that is located well
below spatial and temporal frequencies of the Bragg limit [8] beyond which the
stopband structure starts, where only the lowest s-polarized mode (considered
in the paper) and the lowest p-polarized mode (not considered in the paper)
exist in the medium, as will be done in Sec. V. In this regime, the sinusoidal
modulation profile may be approximated by a bilayer periodic profile, with
layer indices $n_{1}$ and $n_{2}$, and the corresponding configuration may be
seen as an effective “atom - vacuum” sequence that is akin to the atom-vacuum
structure of the real medium in Fig. 1(a).

##  III Methodology and Assumptions

We shall now derive and compare the dispersion relations and Poynting-vector
directions for the two media in Fig. 1. This will be done using some
mathematical tools of general relativity, specifically the Maxwell-Cartan
equations [48], the Rindler transformations and metric [49], and the general
tensorial coordinate transformation formulas [50] (see Appendix A), and making
extensive use of frame hopping between the laboratory frame, $K$ (unprimed
variables), and the moving frame, $K^{\prime}$ (primed variables). We shall
assume that the problem is invariant in the $y$ direction (Fig. 1), and hence
spatially two-dimensional, and restrict our attention to the case of s
polarization, where the electric field is directed along the $y$ direction,
the case of p-polarization being formally analogous.

##  IV Accelerated-Matter Medium

Let us start our electrodynamic study with the case of the accelerated-matter
medium [Fig. 1(a)]. As is usually done in such problems, to avoid the
complexity of bianisotropy, we first compute the dispersion relations and the
fields in the $K^{\prime}$ frame, whose noninertiality requires a general-
relativistic treatment, and then transform these quantities back to the $K$
frame. Writing the $K^{\prime}$-frame dispersion relation in its covariant
form, i.e., $g_{\mu^{\prime}\nu^{\prime}}k^{\mu^{\prime}}k^{\nu^{\prime}}=0$,
where $k^{\mu^{\prime}}=(\omega^{\prime}/c,\mathbf{k^{\prime}})$, and inverse-
Rindler transforming the resulting relation yields the $K$-frame dispersion
relation (see Appendix B.1)

$\frac{\left(k_{z}-\raisebox{2.0pt}{$\chi$}\omega/c\right)^{2}}{\alpha^{2}n^{\prime
2}}+\frac{k_{x}^{2}}{\alpha n^{\prime 2}}=\left(\frac{\omega}{c}\right)^{2}.$
(1)

In this relation, $n^{\prime}$ is the ($K^{\prime}$-frame) refractive index of
the medium, which is assumed to be isotropic, dispersionless and linear so
that $n^{\prime}$ is a constant, $\alpha=(1-\beta^{2})/(1-n^{\prime
2}\beta^{2})$ and $\raisebox{2.0pt}{$\chi$}=\beta(1-n^{\prime 2})/(1-n^{\prime
2}\beta^{2})$, where $\beta=v/c=\tanh(a^{\prime}t^{\prime}/c+\xi_{0})$ is the
instantaneous normalized velocity of the medium [Fig. 1(a)], with $a^{\prime}$
being the (constant) proper acceleration and
$\xi_{0}=\sinh^{-1}(\beta_{0}\gamma_{0})$, where $\beta_{0}=v(0)/c$ and
$\smash{\gamma_{0}=1/\sqrt{1-\beta_{0}^{2}}}$ are the normalized initial
velocity and the corresponding Lorentz factor.

On the other hand, the direction of the Poynting vector, $\mathbf{S}$, may be
obtained by writing the wave equation in the $K^{\prime}$-frame, solving it
using a vacuum-field ansatz, and inverse-Rindler transforming the resulting
field expressions [51], which yields (see Appendix B.2)

$\theta_{\mathbf{S}}=\tan^{-1}\left(\gamma\frac{\sin\theta_{n}^{\prime}+n^{\prime}\beta}{\cos\theta_{n}^{\prime}}\right),$
(2)

where
$\smash{\gamma=1/\sqrt{1-\beta^{2}}=\cosh(a^{\prime}t^{\prime}/c+\xi_{0})}$ is
the instantaneous Lorentz factor and $\sin\theta_{n}^{\prime}=N/D$ with
$N=1-\sin\theta\tanh(a^{\prime}t^{\prime}/cn^{\prime}+\xi_{0})$ and
$D=\sin\theta-\tanh(a^{\prime}t^{\prime}/cn^{\prime}+\xi_{0})$, where $\theta$
is the incident (initial) angle, shown in the top panels of Fig. 1. Note that
Eq. (2), although partially expressed in terms of $K^{\prime}$-frame
variables, for the sake of compactness, really represents a $K$-frame
quantity.

##  V Accelerated-Modulation Medium

Let us move on now to the case of the accelerated-modulation medium, which is
the main object of this paper. The corresponding analysis is considerably more
involved than that of the accelerated-matter medium, due to both the more
complex, bilayer structure of the medium [Fig. 1(b)] and the inexistence of a
motion-less frame, as we shall see next. Here, in the $K$ frame, the
modulation is accelerating and matter is stationary, while in the $K^{\prime}$
frame, the modulation is stationary and matter is accelerating, in the
opposite direction, with velocity $v^{\prime}=-v$. So, motion occurs in both
frames, which is unusual in conventional relativity problems. We still elect
to attack the problem in the $K^{\prime}$ frame, on the ground that moving
matter with stationary boundaries [here, the interfaces between the layers in
Fig. 1(b)] is a known problem [52], where the addition of noninertiality is
tractable using the tools of general relativity.

In the $K^{\prime}$ frame, the two media forming the stratified structure
[Fig. 1(b)], assumed to be isotropic in $K$, with scalar refractive indices,
relative permittivities and relative permeabilities $n_{1,2}$,
$\epsilon_{1,2}$ and $\mu_{1,2}$ ($n_{1,2}=\sqrt{\epsilon_{1,2}\mu_{1,2}}$),
are bianisotropic, due to the motion of their constituent matter, and
noninertial, due to acceleration. Upon this consideration, the sought after
dispersion relation may be obtained by first constructing the corresponding
$K^{\prime}$-frame constitutive relations, which include the tensorial
constitutive parameters
$\smash{\overline{\overline{\epsilon}}_{1,2}^{\prime}}$,
$\smash{\overline{\overline{\mu}}_{1,2}^{\prime}}$ and
$\smash{\overline{\overline{\chi}}_{1,2}^{\prime}}$, then space-time averaging
the tangential and normal components of these parameters (metamaterial
regime), next inverse-Rindler transforming the so-obtained averages, and
finally substituting the resulting expressions into the general ($K$-frame)
bianistropic dispersion relation [46] (see Appendix C.1). This yields

$\frac{\left(k_{z}-\overline{\raisebox{2.0pt}{$\chi$}}\omega/c\right)^{2}}{\overline{\epsilon}_{\|}\overline{\mu}_{\|}}+\frac{k_{x}^{2}}{\overline{\epsilon}_{\|}\overline{\mu}_{\perp}}=\left(\frac{\omega}{c}\right)^{2},$
(3)

which involves the averaged tangential and normal constitutive parameters

$\overline{\epsilon}_{\|}=\frac{\Sigma_{\epsilon}-\beta^{2}\epsilon_{1}\epsilon_{2}\Sigma_{\mu}}{1-\beta^{2}\Sigma_{\epsilon}\Sigma_{\mu}},\quad\overline{\epsilon}_{\perp}=\frac{2\epsilon_{1}\epsilon_{2}}{\Sigma_{\epsilon}},$
(4a)
$\overline{\mu}_{\|}=\frac{\Sigma_{\mu}-\beta^{2}\mu_{1}\mu_{2}\Sigma_{\epsilon}}{1-\beta^{2}\Sigma_{\epsilon}\Sigma_{\mu}},\quad\overline{\mu}_{\perp}=\frac{2\mu_{1}\mu_{2}}{\Sigma_{\mu}}$
(4b) and
$\overline{\raisebox{2.0pt}{$\chi$}}=\beta\frac{\Delta_{\epsilon}\Delta_{\mu}}{1-\beta^{2}\Sigma_{\epsilon}\Sigma_{\mu}},$
(4c) where $\Sigma_{\epsilon}=(\epsilon_{1}+\epsilon_{2})/2$,
$\Delta_{\epsilon}=(\epsilon_{1}-\epsilon_{2})/2$, and similarly for
$\Sigma_{\mu}$ and $\Delta_{\mu}$.

On the other hand, the Poynting vector direction may be obtained by taking the
spatial Fourier transforms of the $K$-frame Maxwell equations and dispersion
relations, inserting the resulting former expression into the resulting latter
expression, eliminating the electric and magnetic flux density fields, and
taking the appropriate ratio of the remaining field components (see Appendix
C.2), which yields

$\theta_{\mathbf{S}}=\tan^{-1}\left({\frac{\overline{\mu}_{\perp}}{\overline{\mu}_{\|}}\frac{\sin\theta-\overline{\raisebox{2.0pt}{$\chi$}}}{\cos\theta}}\right),$
(5)

whose parameters are given by Eqs. (4).

##  VI Formal Modulation-Matter Equivalence

Comparing Eq. (3) with Eq. (1) reveals a most interesting fact: the
accelerated-modulation medium is formally equivalent to the accelerated-matter
medium (in $K$). Both media are indeed bianisotropic, with identical tensorial
structure. Thus, an accelerated-modulation space-time metamaterial can
potentially alter light in the same manner as accelerated matter and hence, by
the equivalence principle, as gravitation.

##  VII Dispersion and Propagation

Figure 2 shows typical temporal evolutions of the isofrequency contours for
the accelerated media of interest. Figure 2(a) corresponds to the accelerated-
matter case, computed by Eq. (1), while Fig. 2(b) corresponds to the
accelerated-modulation case, computed by Eq. (3), respectively. As time
passes, the isofrequency contours progressively shift, in both cases, parallel
to the direction of motion, $z$. In the case of accelerated matter [Fig.
2(a)], the shift occurs in the $-z$ direction, which corresponds to deflection
of energy (direction $\mathbf{v}_{\mathrm{g}}$ in the figure) in the $+z$
direction. This deflection is a manifestation of the Fresnel-Fizeau drag [14,
15], which adds the momentum
$\overline{\raisebox{2.0pt}{$\chi$}}k_{0}\propto\beta$, increasing with time
due acceleration, to the $k_{z}$ component of the wave [Eq. (1)]. The physics
is drastically different in the case of accelerated modulation [Fig. 2(b)].
Now, the isofrequency contour shift is in the $+z$ direction, corresponding to
deflection of energy in the $-z$ direction, i.e., _contra-directionally_ to
the motion [8, 24, 25]. Such contra-directional deflection is _not_ due to the
Fresnel-Fizeau effect, since no motion of matter occurs in the laboratory
frame; it results from the _space-time weighted averaging_ [8] of the
constitutive parameters in Eq. (4), whereby the wave traveling along the
direction of the modulation, spending more time in the denser layers ($n_{1}$,
assuming $n_{1}>n_{2}$), propagates slower in that direction than in the
opposite direction [25] 333This space-time weighted averaging effect was
discovered in [8] and extensively described in [25]. In these papers, the
effect was derived for the case of a _uniformly moving_ modulation, but the
related argument still applies to the case of acceleration because an
accelerated medium may be considered as having locally a uniform velocity at
any given point of space and time [e.g., final time in the inset of Fig.
2(b)], so that the global deflection of the wave energy (global direction of
the beam in Fig. 3) is entirely determined by a single point of space-time.
The deflection of a beam in a uniform-velocity modulation medium was first of
explicitly demonstrated in [24], but with the problematic invocation of the
“Fresnel drag” as an explanation for the effect. The Fresnel-Fizeau [14, 15]
drag implies moving matter (fluid in Fizeau’s original experiment [15]),
whereas no matter (atoms and molecules) is moving in the moving-modulation
medium. One could argue that interfaces move, and that the “Fresnel drag”
would be one related by these moving interfaces instead of moving matter, but
the problem is that this would predict the wrong direction of deflection!,
which imparts the negative momentum contribution
$-\left|\overline{\raisebox{2.0pt}{$\chi$}}\right|k_{0}\propto-\beta$ to the
$k_{z}$ component of the wave [Eq. (3)] 444Note that the isofrequency curve at
$t=0$ (before the onset of the modulation) is _not_ a circle [Fig. 2(b)].
Mathematically, this may be seen by inserting Eq (4) into (3) with $\beta=0$,
which implies $\overline{\raisebox{2.0pt}{$\chi$}}=0$; this operation centers
the curve to the origin (absence of motion) but does not suppress its
eccentricity. The remaining elliptical shape simply corresponds to the
anisotropy of the stratified structure of the modulation medium [see Fig. 1(b)
at $t=0$]..

Figure 2: Temporal evolution of isofrequency curves for the accelerated-(a)
matter medium [Eq. (1)] and (b) modulation medium [Eq. (3) with (4)] in Fig.
1, with proper acceleration $a^{\prime}/c^{2}=0.45$ and initial normalized
velocity $\beta_{0}=0$. The time $t=0$ corresponds to the onset of the
modulation. The modulation medium has the $K$-frame refractive indices
$n_{1}=\sqrt{\epsilon_{1}\mu_{1}}$ and $n_{2}=\sqrt{\epsilon_{2}\mu_{2}}$,
with $\epsilon_{1}=\mu_{1}=1.5$ and $\epsilon_{2}=\mu_{2}=3$, while the matter
medium has the $K^{\prime}$-frame refractive index
$n^{\prime}=\sqrt{\epsilon^{\prime}\mu^{\prime}}$ with
$\smash{\epsilon^{\prime}=\Sigma_{\epsilon}}$ and
$\smash{\mu^{\prime}=\Sigma_{\mu}}$, corresponding to $n^{\prime}=2.25$. The
insets show the isofrequencies $t=0$ and $t=2.2$ ns and corresponding
representative phase velocity vector, $\mathbf{v}_{\mathrm{p}}$, and group
velocity vector, $\mathbf{v}_{\mathrm{g}}$ [with directions given by Eqs. (2)
and (5)].

In order to further examine this contra-directional deflection effect, let us
consider the propagation of a Gaussian beam, launched perpendicularly to the
direction of motion [$x$ direction in Fig. 1(b)] for maximal deflection. Such
a field may be obtained by inserting the paraxial approximation of the
dispersion relation in Eq. (3) into the integral expression for the field
corresponding to the propagating Gaussian beam, and calculating the resulting
integral, which yields (see Appendix C.3)

$\begin{split}E_{y}(z,x)=E_{0}\frac{\text{e}^{\mathrm{i}n_{\|}k_{0}(1-\delta)x}}{1+\mathrm{i}\frac{n_{\|}2x}{n_{\perp}^{2}k_{0}W_{0}^{2}}}\exp\left(-\frac{\left(z+z_{\delta}\right)^{2}}{\left(1+\mathrm{i}\frac{n_{\|}2x}{n_{\perp}^{2}k_{0}W_{0}^{2}}\right)W_{0}^{2}}\right),\end{split}$
(6)

where
$\smash{\overline{n}_{\|}=\sqrt{\overline{\epsilon}_{\|}\overline{\mu}_{\perp}}}$,
$\smash{\overline{n}_{\perp}=\sqrt{\overline{\epsilon}_{\|}\overline{\mu}_{\|}}}$,
$\smash{\delta=\overline{\raisebox{2.0pt}{$\chi$}}^{2}k_{0}/2\overline{n}_{\perp}^{2}}$
and
$\smash{z_{\delta}=x\overline{\raisebox{2.0pt}{$\chi$}}\overline{n}_{\|}/\overline{n}_{\perp}^{2}}$
[Eqs. (4)], with $k_{0}=\omega/c=2\pi/\lambda_{0}$, and where $W_{0}$ is the
waist of the beam at $x=0$ 555Note that all the formulas given in the paper
are complicated functions of the odd parameter $\beta$ such that the related
quantities and relations are nonreciprocal..

Figure 3 plots the Gaussian-beam field, computed by Eq. (6), with Figs. 3(a)
and 3(b) corresponding to positive modulation acceleration and negative
modulation acceleration (or deceleration), respectively. A first observation
is that the beam, in both cases, is deflected away from the launching axis
($x$), i.e., in the opposite direction of the velocity ($\beta$), as expected
from the previous results [Fig. 2(b)]. However, the figure also reveals that
the beam is _bent_ by the medium, with bending occurring _towards the
direction of the acceleration_ ($a$). This result indicates that _accelerated
modulation curves space-time_ for light, similarly to gravitation [56, 40,
41]. In fact, such curving could have been inferred from the (known) existence
of deflection (without curving) in a uniform-velocity (or acceleration-less)
modulation medium [8, 24, 25] [Eq. (3) with
$\overline{\raisebox{2.0pt}{$\chi$}}=\mathrm{const.}$ from
$\beta(a^{\prime}=0)=\beta_{0}=\mathrm{const.}$, corresponding to a straight,
vertical elliptic cylinder in Fig. 2(b)]. Indeed, since acceleration is
locally equivalent to uniform velocity and since uniform-velocity sections
with different velocities at different positions of space-time point to
different directions, combining the related infinitesimal uniform sections
automatically produces the observed curving. However, the exact dispersion and
field quantities given above could naturally not have been obtained without a
rigorous general relativistic treatment.

Figure 3: Deflection and bending of a Gaussian beam in the accelerated-
modulation medium in Fig. 1(b), specifically
$\left|\mathrm{Re}\left\\{E_{y}\right\\}\right|$ [Eq. (6)], with the medium
parameters in Fig. 2 ($\epsilon_{1}=\mu_{1}=1.5$ and
$\epsilon_{2}=\mu_{2}=3$), $W_{0}=1.5\lambda_{0}=0.45$ mm and
$|a^{\prime}|/c^{2}=0.45$, for the cases of (a) positive acceleration, with
$\beta_{0}=0$, and (b) negative acceleration (or deceleration), with
$\beta_{0}=0.3$. The arrows associated with $a$ and $\beta$ correspond to the
direction of the acceleration and velocity, respectively.

##  VIII Light Bending for Various Constituent Media

So far, we have implicitly assumed that the two media forming the unit cell of
the space-time modulation metamaterial in Fig. 1(b) have specific double-
positive constitutive parameters ($\epsilon_{1},\mu_{1}>1$ and
$\epsilon_{2},\mu_{2}>1$). However, modulation parameters might take various
and even negative values. Therefore, we shall now perform a parametric
analysis of the light bending effect for different constitutive parameters,
and compare the results with those of moving matter with equivalent average
parameters. This analysis is presented in Fig. 4, which plots the temporal
evolution of the Poynting vector direction for a wave launched in the $x$
direction, with Figs. 4(a) and (b) respectively corresponding to different
double-positive and double-negative medium parameters.

Figure 4: Direction of the Poynting vector (angle shown in Figs. 2 and 3)
versus time for different values of the parameter
$\Lambda=\epsilon_{2}/\epsilon_{1}=\mu_{2}/\mu_{1}$ (with
$\smash{\sqrt{\mu_{1}/\epsilon_{1}}=\sqrt{\mu_{2}/\epsilon_{2}}}$ for local
matching) in the accelerated-modulation medium [Eq. (5)] and in the
accelerated-matter medium [Eq. (2)] for $a^{\prime}/c^{2}=0.83$ (and hence
$a>0$) and $\beta_{0}=0$ (and hence $v>0$). The index $n^{\prime}$ for matter
has been chosen as in Fig. 2 for proper Fresnel-Fizeau density and
quantitative comparison. (a) Double-positive ($n^{\prime 2}>0$) and (b)
double-negative ($n^{\prime 2}>0$) medium parameters.

In the case of matter, all the results in Fig. 4 (dashed curves) may be simply
explained in terms of the Fresnel-Fizeau drag effect, according to which the
velocity of light in the moving-matter medium is given by
$v_{\text{light}}=c/n^{\prime}+v(1-1/n^{\prime 2})$ [15], where $n^{\prime}$
is the refractive index of the matter medium at rest. Since it occurs in the
$z$ direction, the drag effect implies $\theta_{\mathbf{S}}\propto
v(1-1/n^{\prime 2})$. It may be easily checked that results in the figure
precisely follow this prediction for the two different types of medium
constituents, with the factor $v$ ($\partial v/\partial t=a>0$) accounting for
the temporal evolution of any given curve and factor $1-1/n^{\prime 2}$
accounting the signs and difference between the different curves at any given
time.

Let us now examine the results for the modulation case (solid curves in Fig.
4). The figure shows that the metamaterial can reach a great range of energy
deflection and bending amounts upon proper parametric tuning, similarly to its
matter medium counterpart, as expected from the formal equivalence between the
two types of media that was previously pointed out. These results are harder
to explain than those for matter, due to the greater complexity of the
metamaterial structure and related space-time averaged quantities [Eqs. (4)].
However, the negative direction of deflection and bending for the double-
positive constituent-media structure [Figs. 4(a)] clearly corresponds to the
motion-contradirectional effect that was explained in connection with Fig.
2(b), while the strength of the effect is explained by the amount of contrast
between the layer parameters, $\Delta_{\epsilon,\mu}$, since it is this
contrast that forms the medium. Moreover, the results for the double-negative
constituent-media structure [Fig. 4(b)] may be understood as follows. For
$\Lambda=-1$, the structure alternates equal-magnitude positive and negative
index layers so as to form a Pendry lens [57] periodic configuration [58],
here with moving interfaces, where the positive and negative $x$ momentum
components across the unit cell cancel out so as to produce purely
$z$-directed propagation. For $\Lambda\neq-1$, the refractive indices of the
two layers of the unit cell have different magnitudes, which breaks the
$\Lambda=-1$ $x$-momentum anti-symmetry; as a result, the rays in the two
layers undergo different deflections, which produces a net wave deflection,
proportional to the index contrast.

##  IX Analog-Gravity Media

While the paper has focused onto a uniform-direction and constant-proper
acceleration profile until this point, the reported accelerated-modulation
space-time metamaterials may assume a virtually unlimited diversity of
metrics, and potentially feature equally diverse opportunities for light
manipulation. To illustrate this, Fig. 5 presents metamaterials that mimic
Schwarzschild holes [59], based on an average refractive index profile that
follows the corresponding metric [60]. The black hole, shown in Fig. 5,
displays the well-know light attraction and absorption effects, while the
white hole, whose cosmological existence is purely theoretical but which may
actually be implemented by an accelerated-modulation metamaterial, displays
the complementary light deviation and repelling effects (see Appendix D).

Figure 5: Scattering of a Gaussian beam of light by Schwarzschild (a) black
and (b) white holes based on a cylindrical version of the accelerated bilayer-
unit-cell space-time modulation medium in Fig. 1(b)
($\left|\mathrm{Re}\left\\{E_{y}\right\\}\right|$ Appendix D), for offset (top
panels) and facing (bottom panels) illuminations, with
$r_{\mathrm{s}}=3.15\lambda_{0}=0.63$ mm (horizon radius), and
$W_{0}=6.5\lambda_{0}$ for (a) and $W_{0}=8.5\lambda_{0}$ for (b).

##  X Conclusion

This paper has initiated the field of accelerated-modulation space-time
metamaterials. The capability of such materials to curve space-time of light,
as gravitation does, augurs vast opportunities for scientific and
technological development with multiple applications in electronics,
electromagnetics and optics.

##  Appendix A General Relativity Tools

The key results in the paper are obtained using general relativistic
electrodynamics, specifically the Maxwell-Cartan equations, the Rindler
transformations of the space-time variables and the general tensorial
coordinate transformation formulas.

The Maxwell-Cartan equations are generalizations of the Maxwell’s equations
for curved space-time. They read [48, 42]

$\partial_{(\lambda}F_{\mu\nu)}=\partial_{\lambda}F_{\mu\nu}+\partial_{\nu}F_{\lambda\mu}+\partial_{\mu}F_{\nu\lambda}=0$
(7a) and $\partial_{\mu}G^{\mu\nu}=0,$ (7b)

where $F_{\mu\nu}$ is the Faraday tensor and $G^{\mu\nu}$ is its dual [61],
which are defined as

$F_{\mu\nu}=\begin{bmatrix}0&E_{x}&E_{y}&E_{z}\\\ -E_{x}&0&-cB_{z}&cB_{y}\\\
-E_{y}&cB_{z}&0&-cB_{x}\\\ -E_{z}&-cB_{y}&cB_{x}&0\end{bmatrix}$ (8a) and
$G^{\mu\nu}=\begin{bmatrix}0&-cD_{x}&-cD_{y}&-cD_{z}\\\
cD_{x}&0&H_{z}&-H_{y}\\\ cD_{y}&-H_{z}&0&H_{x}\\\
cD_{z}&H_{y}&-H_{x}&0\end{bmatrix}.$ (8b)

Note that the Einstein summation convention, whereby repeated indices imply
summation over corresponding indices, are assumed everywhere in the paper, as
for instance in (7b), where
$\partial_{\mu}G^{\mu\nu}=\sum_{\mu=0}^{3}\partial_{\mu}G^{\mu\nu}$.

The Rindler transformations are the coordinate transformations between the
laboratory frame and the moving frame for the case of constant proper
acceleration [49]. We shall use here the Kottler-Møller [62] version of the
Rindler transformations [49] to accommodate nonzero constant initial
velocities. The Kottler-Møller transformations, assuming propagation in the
$z$-direction, are given by the relations

$ct=\frac{c^{2}}{a^{\prime}}\sqrt{g_{00}}\sinh(\xi+\xi_{0})-\frac{c^{2}}{a^{\prime}}\sinh(\xi_{0}),$
(9a) $x=x^{\prime},$ (9b) $~{}y=y^{\prime}$ (9c) and
$z=\frac{c^{2}}{a^{\prime}}\sqrt{g_{00}}\cosh(\xi+\xi_{0})-\frac{c^{2}}{a^{\prime}}\cosh(\xi_{0}),$
(9d) where $\xi=a^{\prime}t^{\prime}/c$ (9e) and
$\xi_{0}=\sinh^{-1}(\beta_{0}\gamma_{0}),$ (9f)

with $a^{\prime}$ being the (constant) proper acceleration, $\beta_{0}$ and
$\gamma_{0}$ being the initial relative velocity and Lorentz factor,
respectively, and where

$g_{00}=\left(1+(a^{\prime}z^{\prime}/c^{2})\right)^{2},$ (10a)

is the $00$-term of the Rindler metric, which is simply
$g_{\mu^{\prime}\nu^{\prime}}=\mathrm{diag}(g_{00},1,1,1)$. The corresponding
($K$-frame) instantaneous normalized velocity may be calculated by dividing
(9d) and (9a) by $dt^{\prime}$, calculating the related derivatives and taking
the ratio of the resulting expressions, which yields

$\beta=\frac{dz}{d(ct)}=\frac{dz/d(ct^{\prime})}{d(ct)/d(ct^{\prime})}=\tanh(\xi+\xi_{0}),$
(11)

while the ($K$-frame) acceleration may be derived by taking the time
derivative of this equation, which gives

$a=a^{\prime}\left(\cosh(\xi+\xi_{0})\right)^{-3}.$ (12)

The inverse of (9a) and (9d) may be found as follows. First, we isolate
$\sinh(\xi+\xi_{0})$ in (9a) and $\cosh(\xi+\xi_{0})$ in (9d). Then, we obtain
the relation $t^{\prime}=t^{\prime}(t)$ by successively taking the ratio of
the resulting expressions and the inverse of the new expression, while we
obtain the relation $z^{\prime}=z^{\prime}(z)$ by squaring the resulting
expressions, taking the difference of the new expressions and taking the
square root of the result. The final result is

$ct^{\prime}=\frac{c^{2}}{a^{\prime}}\tanh^{-1}\left(\frac{ct+\frac{c^{2}}{a^{\prime}}\sinh(\xi_{0})}{z+\frac{c^{2}}{a^{\prime}}\cosh(\xi_{0})}\right)-c\xi_{0}$
(13a) and
$z^{\prime}=\sqrt{(z+\frac{c^{2}}{a^{\prime}}\cosh(\xi_{0}))^{2}-(ct+\frac{c^{2}}{a^{\prime}}\sinh(\xi_{0}))^{2}}-\frac{c^{2}}{a^{\prime}}.$
(13b)

Finally, the tensorial coordinate transformations of the fields are given by
the covariant and contravariant relations [50]:

$F_{\mu^{\prime}\nu^{\prime}}=\frac{\partial x^{\rho}}{\partial
x^{\mu^{\prime}}}\frac{\partial x^{\sigma}}{\partial
x^{\nu^{\prime}}}F_{\rho\sigma}$ (14a) and
$\left|\text{det}\left(\frac{\partial x^{\mu^{\prime}}}{\partial
x^{\nu^{\prime}}}\right)\right|G^{\mu^{\prime}\nu^{\prime}}=\frac{\partial
x^{\mu^{\prime}}}{\partial x^{\rho}}\frac{\partial x^{\nu^{\prime}}}{\partial
x^{\sigma}}G^{\rho\sigma}.$ (14b)

For a rank-one tensor, such as the wave tensor ($k_{\mu}$), these relations
reduce to

$k_{\mu^{\prime}}=\frac{\partial x^{\rho}}{\partial x^{\mu^{\prime}}}k_{\rho}$
(15a) and $k^{\mu^{\prime}}=\frac{\partial x^{\mu^{\prime}}}{\partial
x^{\rho}}k^{\rho}.$ (15b)

##  Appendix B Accelerated-Matter Media

###  B.1 Dispersion Relation

In the $K^{\prime}$-frame, matter is stationary and isotropic. However,
$K^{\prime}$ is noninertial, which requires a general relativistic treatment
[42]. In particular, the dispersion relation must be written in its covariant
form [63], viz.,

$k_{\mu^{\prime}}k^{\mu^{\prime}}=0,$ (16)

where

$k^{\mu^{\prime}}=\left(\frac{\omega^{\prime}}{c},\mathbf{k^{\prime}}/n^{\prime}\right)$
(17)

is the contravariant form of the wave tensor, with $\mathbf{k}^{\prime}$ being
the wave vector, and the covariant form of the wave tensor is found by
contraction with the metric tensor, which yields

$k_{\mu^{\prime}}=g_{\mu^{\prime}\nu^{\prime}}k^{\nu^{\prime}}=\left(-g_{00}\frac{\omega^{\prime}}{c},\mathbf{k^{\prime}}/n^{\prime}\right).$
(18)

Inserting then (17) and (18) into (16) yields the $K^{\prime}$-frame
dispersion relation

$\frac{k^{\prime 2}_{z}}{n^{\prime 2}}+\frac{k^{\prime 2}_{x}}{n^{\prime
2}}=g_{00}\left(\frac{\omega^{\prime}}{c}\right)^{2}.$ (19)

Finally, inverse-transforming (19) with (9a) to (9d) according to (15) leads
to the $K$-frame dispersion relation

$\frac{(k_{z}-\raisebox{2.0pt}{$\chi$}\omega/c)^{2}}{\alpha^{2}n^{\prime
2}}+\frac{k_{x}^{2}}{\alpha n^{\prime 2}}=\left(\frac{\omega}{c}\right)^{2},$
(20a) where $\alpha=(1-\beta^{2})/(1-n^{\prime 2}\beta^{2})$ (20b) and
$\raisebox{2.0pt}{$\chi$}=\beta(1-n^{\prime 2})/(1-n^{\prime 2}\beta^{2}),$
(20c)

with $n^{\prime}$ being the refractive index of the medium measured in the
$K^{\prime}$-frame.

###  B.2 Fields and Deflection Angles

Let us now consider an incident plane wave in vacuum, which will be later used
as an ansatz to calculate the fields of interest. Under the assumption of s
polarization, depicted in Fig. 6, such a wave satisfies the relations
$E_{x}=E_{z}=B_{y}=D_{x}=D_{z}=H_{y}=0$, and its Faraday tensor and dual
Faraday tensor are then found, upon inserting these relations into Eqs. (8),
as

$F_{\mu\nu}=\begin{bmatrix}0&0&1&0\\\ 0&0&-\cos\theta&0\\\
-1&\cos\theta&0&\sin\theta\\\
0&0&-\sin\theta&0\end{bmatrix}E_{0}f\left(\zeta\right)$ (21a) and
$G^{\mu\nu}=\begin{bmatrix}0&0&-\eta_{0}^{-1}&0\\\
0&0&\frac{\cos\theta}{\eta_{0}}&0\\\
\eta_{0}^{-1}&-\frac{\cos\theta}{\eta_{0}}&0&\frac{\sin\theta}{\eta_{0}}\\\
0&0&-\frac{\sin\theta}{\eta_{0}}&0\end{bmatrix}E_{0}f\left(\zeta\right),$
(21b) after defining $\zeta=z\sin{\theta}+x\cos{\theta}-ct,$ (21c)

where $f(\cdot)$ and $\theta$ are the incident (initial) wave profile and
angle, respectively.

Figure 6: Incident and propagating fields configuration.

The fields in the $K$-frame may be derived by solving the wave equation in the
$K^{\prime}$-frame and inverse-Rindler transforming the results to the
$K$-frame, which results in [51]

$\begin{split}&F_{\mu\nu}=\gamma\frac{P_{n}}{n^{\prime}}E_{0}f(\zeta)\\\
&\begin{bmatrix}0&0&1+n^{\prime}\beta\sin\theta^{\prime}_{n}&0\\\
0&0&\frac{n^{\prime}\cos{\theta^{\prime}_{n}}}{\gamma}&0\\\
-(1+n^{\prime}\beta\sin\theta^{\prime}_{n})&-\frac{n^{\prime}\cos{\theta^{\prime}_{n}}}{\gamma}&0&(n^{\prime}\sin\theta^{\prime}_{n}+\beta)\\\
0&0&-(n^{\prime}\sin\theta^{\prime}_{n}+\beta)&0\end{bmatrix}\end{split}$
(22a) and $\begin{split}&G^{\mu\nu}=\gamma P_{n}E_{0}f(\zeta)\\\
&\begin{bmatrix}0&0&-\frac{n^{\prime}+\beta\sin\theta^{\prime}_{n}}{\eta_{0}\mu^{\prime}}&0\\\
0&0&\frac{\cos\theta^{\prime}_{n}}{\gamma\eta_{0}\mu^{\prime}}&0\\\
\frac{n^{\prime}+\beta\sin\theta^{\prime}_{n}}{\eta_{0}\mu^{\prime}}&-\frac{\cos\theta^{\prime}_{n}}{\gamma\eta_{0}\mu^{\prime}}&0&-\frac{\sin\theta^{\prime}_{n}+n^{\prime}\beta}{\eta_{0}\mu^{\prime}}\\\
0&0&\frac{\sin\theta^{\prime}_{n}+n^{\prime}\beta}{\eta_{0}\mu^{\prime}}&0\end{bmatrix},\end{split}$
(22b) where
$P_{n}=\cosh(\xi/n^{\prime}+\xi_{0})-\sin(\theta)\sinh(\xi/n^{\prime}+\xi_{0}),$
(22c) $\cos\theta^{\prime}_{n}=\frac{\cos\theta}{P_{n}}$ (22d) and
$\gamma=1/\sqrt{1-\beta^{2}}=\cosh(\xi+\xi_{0}),$ (22e)

with $\beta$ and $n^{\prime}=\sqrt{\epsilon^{\prime}\mu^{\prime}}$ being the
instantaneous velocity and the stationary refractive index of the medium.

The Poynting vector is given by

$\mathbf{S}=\mathbf{E}\times\mathbf{H}=-E_{y}H_{x}\hat{z}+E_{y}H_{z}\hat{x},$
(23)

and its angle with respect to the $x$-axis can be written as

$\theta_{\mathbf{S}}=-\tan^{-1}\left(\frac{E_{y}H_{x}}{E_{y}H_{z}}\right)=-\tan^{-1}\left(\frac{H_{x}}{H_{z}}\right),$
(24)

where $H_{z}$ and $H_{x}$ are found in (22b) with the help of (8b), which
results into

$\theta_{\mathbf{S}}=\tan^{-1}\left(\gamma\frac{\sin\theta_{n}^{\prime}+n^{\prime}\beta}{\cos\theta_{n}^{\prime}}\right).$
(25)

where $\theta^{\prime}_{n}$ is given in (22d).

The deflection angle is the angle the Poynting vector makes with the phase
angle ($\theta_{\mathbf{k}}=\tan^{-1}(k_{z}/k_{x})$). Therefore, it can be
represented as
$\theta_{\mathrm{defl}}=\theta_{\mathbf{S}}-\theta_{\mathbf{k}}$, and
explicitly written

$\theta_{\mathrm{defl}}=\tan^{-1}{\left(\frac{\gamma(Q_{n}+n^{\prime}\beta
P_{n})-\sin\theta}{\cos\theta+\gamma(Q_{n}+n^{\prime}\beta
P_{n})\tan\theta}\right)},$ (26)

where

$Q_{n}=\sin\theta\cosh(\xi_{n}^{\prime}+\xi_{0})-\sinh(\xi_{n}^{\prime}+\xi_{0}),$
(27)

and $P_{n}$ is given in (22c).

##  Appendix C Accelerated-Modulation Media

###  C.1 Dispersion Relation

Under the assumption of (moving-_modulation_) layer isotropy in the $K$-frame,
the $K$-frame constitutive relations read

$\mathbf{B}_{1,2}=\mu_{1,2}\mathbf{H}_{1,2}$ (28a) and
$\mathbf{D}_{1,2}=\epsilon_{1,2}\mathbf{E}_{1,2}.$ (28b)

Rindler transforming (28) to the $K^{\prime}$-frame using (14) with (9)
yields, for each of the two layers, the bianisotropic constitutive relations

$\sqrt{g_{00}}\mathbf{B}^{\prime}_{1,2}=\mu_{0}\overline{\overline{\mu}}^{\prime}_{1,2}\mathbf{H^{\prime}}_{1,2}+\overline{\overline{\raisebox{2.0pt}{$\chi$}}}^{\prime}_{1,2}\mathbf{E}^{\prime}_{1,2}$
(29a) and
$\sqrt{g_{00}}\mathbf{D}^{\prime}_{1,2}=\epsilon_{0}\overline{\overline{\epsilon}}^{\prime}_{1,2}\mathbf{E}^{\prime}_{1,2}-\overline{\overline{\raisebox{2.0pt}{$\chi$}}}^{\prime}_{1,2}\mathbf{H}^{\prime}_{1,2},$
(29b)

with

$\overline{\overline{\mu}}^{\prime}_{1,2}=\begin{bmatrix}\alpha^{\prime}_{1,2}\mu^{\prime}_{1,2}&0&0\\\
0&\alpha^{\prime}_{1,2}\mu^{\prime}_{1,2}&0\\\
0&0&\mu^{\prime}_{1,2}\end{bmatrix},$ (30a)
$\overline{\overline{\epsilon}}^{\prime}_{1,2}=\begin{bmatrix}\alpha^{\prime}_{1,2}\epsilon^{\prime}_{1,2}&0&0\\\
0&\alpha^{\prime}_{1,2}\epsilon^{\prime}_{1,2}&0\\\
0&0&\epsilon^{\prime}_{1,2}\end{bmatrix}$ (30b) and
$\overline{\overline{\raisebox{2.0pt}{$\chi$}}}^{\prime}_{1,2}=\begin{bmatrix}0&\raisebox{2.0pt}{$\chi$}^{\prime}_{1,2}/c&0\\\
-\raisebox{2.0pt}{$\chi$}^{\prime}_{1,2}/c&0&0\\\ 0&0&0\end{bmatrix},$ (30c)
where
$\alpha^{\prime}_{1,2}=(1-\beta^{2})/(1-\epsilon^{\prime}_{1,2}\mu^{\prime}_{1,2}\beta^{2})$
(30d) and
$\raisebox{2.0pt}{$\chi$}^{\prime}_{1,2}=-\beta(1-\epsilon^{\prime}_{1,2}\mu^{\prime}_{1,2})/(1-\epsilon^{\prime}_{1,2}\mu^{\prime}_{1,2}\beta^{2}).$
(30e)

Averaging these relations via the continuity of the field
$\mathbf{E}^{\prime}_{\|}$, $\mathbf{H}^{\prime}_{\|}$, $D^{\prime}_{\perp}$
and $B^{\prime}_{\perp}$ at the layer interfaces yields the homogenized
constitutive relations

$\begin{split}\sqrt{g_{00}}\begin{bmatrix}B^{\prime}_{x}\\\ B^{\prime}_{y}\\\
B^{\prime}_{z}\end{bmatrix}=&\mu_{0}\begin{bmatrix}\overline{\mu}^{\prime}_{\|}&0&0\\\
0&\overline{\mu}^{\prime}_{\|}&0\\\
0&0&\overline{\mu}^{\prime}_{\perp}\end{bmatrix}\begin{bmatrix}H^{\prime}_{x}\\\
H^{\prime}_{y}\\\ H^{\prime}_{z}\end{bmatrix}\\\
&+\begin{bmatrix}0&\overline{\raisebox{2.0pt}{$\chi$}^{\prime}}/c&0\\\
-\overline{\raisebox{2.0pt}{$\chi$}^{\prime}}/c&0&0\\\
0&0&0\end{bmatrix}\begin{bmatrix}E^{\prime}_{x}\\\ E^{\prime}_{y}\\\
E^{\prime}_{z}\end{bmatrix}\end{split}$ (31a) and
$\begin{split}\sqrt{g_{00}}\begin{bmatrix}D^{\prime}_{x}\\\ D^{\prime}_{y}\\\
D^{\prime}_{z}\end{bmatrix}=&\epsilon_{0}\begin{bmatrix}\overline{\epsilon}^{\prime}_{\|}&0&0\\\
0&\overline{\epsilon}^{\prime}_{\|}&0\\\
0&0&\overline{\epsilon}^{\prime}_{\perp}\end{bmatrix}\begin{bmatrix}E^{\prime}_{x}\\\
E^{\prime}_{y}\\\ E^{\prime}_{z}\end{bmatrix}\\\
&+\begin{bmatrix}0&-\overline{\raisebox{2.0pt}{$\chi$}^{\prime}}/c&0\\\
\overline{\raisebox{2.0pt}{$\chi$}^{\prime}}/c&0&0\\\
0&0&0\end{bmatrix}\begin{bmatrix}H^{\prime}_{x}\\\ H^{\prime}_{y}\\\
H^{\prime}_{z}\end{bmatrix},\end{split}$ (31b)

where

$\overline{\epsilon}_{\|}^{\prime}=(\alpha^{\prime}_{1}\epsilon_{1}+\alpha^{\prime}_{2}\epsilon_{2})/2,$
(32a)
$\overline{\epsilon}_{\perp}^{\prime}=2(\epsilon_{1}^{-1}+\epsilon_{2}^{-1})^{-1},$
(32b)
$\overline{\mu}_{\|}^{\prime}=(\alpha^{\prime}_{1}\mu_{1}+\alpha^{\prime}_{2}\mu_{2})/2,$
(32c) $\overline{\mu}_{\perp}^{\prime}=2(\mu_{1}^{-1}+\mu_{2}^{-1})^{-1}$
(32d) and
$\overline{\raisebox{2.0pt}{$\chi$}^{\prime}}=(\raisebox{2.0pt}{$\chi$}^{\prime}_{1}+\raisebox{2.0pt}{$\chi$}^{\prime}_{2})/2.$
(32e)

Following its homogenization, the bilayer-unit-cell moving-modulation medium
at hand has become a bulk medium, whose general dispersion relation is well-
known and is found by inserting the bulk constitutive relations into the wave
equation and taking the Fourier transform of the resulting expression [46].
This relation reads

$\left|\omega^{2}\overline{\overline{\epsilon}}+[\overline{\overline{k}}+\omega\overline{\overline{\xi}}]\cdot\overline{\overline{\mu}}^{-1}\cdot[\overline{\overline{k}}-\omega\overline{\overline{\zeta}}]\right|=0.$
(33)

Inverse-Rindler transforming (31a) and (31b) to the $K$-frame using (14) with
(9) and inserting the resulting equation into (33) yields

$\frac{\left(k_{z}-\overline{\raisebox{2.0pt}{$\chi$}}\omega/c\right)^{2}}{\overline{\epsilon}_{\|}\,\overline{\mu}_{\|}}+\frac{k_{x}^{2}}{\overline{\epsilon}_{\|}\overline{\mu}_{\perp}}=\left(\frac{\omega}{c}\right)^{2},$
(34)

where

$\overline{\epsilon}_{\|}=\overline{\epsilon}_{\|}^{\prime}\frac{(1-\beta^{2})}{(1-\beta\overline{\raisebox{2.0pt}{$\chi$}^{\prime}})^{2}-\overline{\epsilon}_{\|}^{\prime}\,\overline{\mu}_{\|}^{\prime}c^{2}\beta^{2})},$
(35a)
$\overline{\epsilon}_{\perp}=2(\epsilon_{1}^{-1}+\epsilon_{2}^{-1})^{-1},$
(35b)
$\overline{\mu}_{\|}=\overline{\mu}_{\|}^{\prime}\frac{1-\beta^{2}}{(1-\overline{\raisebox{2.0pt}{$\chi$}^{\prime}}\beta)^{2}-\overline{\epsilon}_{\|}^{\prime}\,\overline{\mu}_{\|}^{\prime}c^{2}\beta^{2}},$
(35c) $\overline{\mu}_{\perp}=2(\mu_{1}^{-1}+\mu_{2}^{-1})^{-1}$ (35d) and
$\overline{\raisebox{2.0pt}{$\chi$}}=\frac{((\overline{\raisebox{2.0pt}{$\chi$}^{\prime}}-\beta)(1-\overline{\raisebox{2.0pt}{$\chi$}^{\prime}}\beta)+\epsilon_{\|}^{\prime}\,\mu_{\|}^{\prime}c^{2}\beta)}{((1-\overline{\raisebox{2.0pt}{$\chi$}^{\prime}}\beta)^{2}-\epsilon_{\|}^{\prime}\,\mu_{\|}^{\prime}c^{2}\beta^{2})}.$
(35e)

We can simplify (35) by substituting (32), which leads to

$\overline{\epsilon}_{\|}=\frac{\Sigma_{\epsilon}-\beta^{2}\epsilon_{1}\epsilon_{2}\Sigma_{\mu}}{1-\beta^{2}\Sigma_{\epsilon}\Sigma_{\mu}},$
(36a) $\epsilon_{\perp}=\frac{2\epsilon_{1}\epsilon_{2}}{\Sigma_{\epsilon}},$
(36b)
$\overline{\mu}_{\|}=\frac{\Sigma_{\mu}-\beta^{2}\mu_{1}\mu_{2}\Sigma_{\epsilon}}{1-\beta^{2}\Sigma_{\epsilon}\Sigma_{\mu}},$
(36c) $\mu_{\perp}=\frac{2\mu_{1}\mu_{2}}{\Sigma_{\mu}}$ (36d) and
$\overline{\raisebox{2.0pt}{$\chi$}}=\beta\frac{\Delta_{\epsilon}\Delta_{\mu}}{1-\beta^{2}\Sigma_{\epsilon}\Sigma_{\mu}},$
(36e)

where $\Sigma_{\epsilon}=(\epsilon_{1}+\epsilon_{2})/2$,
$\Delta_{\epsilon}=(\epsilon_{1}-\epsilon_{2})/2$ and similarly for
$\Sigma_{\mu}$, $\Delta_{\mu}$.

###  C.2 Deflection Angles

The fields are solutions to the equations (7), which, upon substitution of
(8), take the explicit form

$\varepsilon^{ijk}\partial_{j}E_{k}=-\frac{\partial}{\partial t}B^{i}$ (37a)
and $\varepsilon^{ijk}\partial_{j}H_{k}=\frac{\partial}{\partial t}D^{i},$
(37b)

where $\varepsilon^{ijk}$ (not to be confused with the permittivity tensor,
$\epsilon^{ij}$) is the Levi-Civita pseudo-tensor [64] and where the Latin
indices run from 1 to 3 (instead of 0 to 3 for the Greek indices in what
precedes).

On the other hand, the constitutive relations may be generally expressed in
the bianisotropic form [46]

$\tilde{D}^{i}=\epsilon^{ij}\tilde{E}_{j}+\varepsilon^{ijk}\raisebox{2.0pt}{$\chi$}_{j}\tilde{H}_{k}$
(38a) and
$\tilde{B}^{i}=\mu^{ij}\tilde{H}_{j}+\varepsilon^{jik}\raisebox{2.0pt}{$\chi$}_{j}\tilde{E}_{k},$
(38b)

where $\epsilon^{ij}$ and $\mu^{ij}$ are the diagonal tensors

$\epsilon^{ij}=\epsilon_{0}\,\mathrm{diag}(\epsilon_{\|},\epsilon_{\|},\epsilon_{\perp})$
(39a) and $\mu^{ij}=\mu_{0}\,\mathrm{diag}(\mu_{\|},\mu_{\|},\mu_{\perp}),$
(39b) and where
$\raisebox{2.0pt}{$\chi$}_{j}=(0,0,\overline{\raisebox{2.0pt}{$\chi$}}/c),$
(39c)

with the different tensor components being given by (35a) to (35e).

In order to combine (37) and (38), we write (37) in the Fourier domain, viz.,

$\varepsilon^{ijk}k_{j}\tilde{E}_{k}=\omega\tilde{B}^{i}$ (40a) and
$\varepsilon^{ijk}k_{j}\tilde{H}_{k}=-\omega\tilde{D}^{i}.$ (40b)

Substituting these relations into (38), and separating the $\tilde{E}_{j}$ and
$\tilde{H}_{j}$ in each of the resulting equations yields

$\epsilon^{ij}\tilde{E}_{j}=-\frac{1}{\omega}\left(\varepsilon^{ijk}k_{j}+\omega\varepsilon^{ijk}\raisebox{2.0pt}{$\chi$}_{j}\right)\tilde{H}_{k}$
(41a) and
$\mu^{ij}\tilde{H}_{j}=\frac{1}{\omega}\left(\varepsilon^{ijk}k_{j}-\omega\varepsilon^{jik}\raisebox{2.0pt}{$\chi$}_{j}\right)\tilde{E}_{k}.$
(41b)

Inserting (35a) to (35e) into (39), and substituting the resulting expressions
into (41) yields

$\tilde{E}_{y}=-\frac{k_{z}-\omega\overline{\raisebox{2.0pt}{$\chi$}}/c}{\omega\epsilon_{\|}}\tilde{H}_{x}+\frac{k_{x}}{\omega\epsilon_{\|}}\tilde{H}_{z},$
(42a)
$\tilde{H}_{x}=-\frac{k_{z}-\omega\overline{\raisebox{2.0pt}{$\chi$}}/c}{\omega\mu_{\|}}\tilde{E}_{y}$
(42b) and $\tilde{H}_{z}=\frac{k_{x}}{\omega\mu_{\perp}}\tilde{E}_{y}.$ (42c)

The $H_{x}$ and $H_{z}$ in the direct domain are then found as the inverse-
Fourier transform of (42b) and(42c), respectively.

The spectral Poynting vector is given by

$\mathbf{\tilde{S}}=\mathbf{\tilde{E}}\times\mathbf{\tilde{H}}=-\tilde{E_{y}}\tilde{H_{x}}\hat{z}+\tilde{E_{y}}\tilde{H_{z}}\hat{x},$
(43)

and its angle with respect to the $x$-axis can then be written as

$\theta_{\mathbf{S}}=-\tan^{-1}\left(\frac{\tilde{E_{y}}\tilde{H}_{x}}{\tilde{E}_{y}\tilde{H}_{z}}\right)=-\tan^{-1}\left(\frac{\tilde{H}_{x}}{\tilde{H}_{z}}\right).$
(44)

Substituting (42b) and (42c) into this relation yields

$\theta_{\mathbf{S}}=\tan^{-1}\left({\frac{\overline{\mu}_{\perp}}{\overline{\mu}_{\|}}\frac{k_{z}-\omega\overline{\raisebox{2.0pt}{$\chi$}}/c}{k_{x}}}\right),$
(45)

which, using $k_{x}=k_{0}\cos\theta$ and $k_{z}=k_{0}\sin\theta$, may also be
written as

$\theta_{\mathbf{S}}=\tan^{-1}\left({\frac{\overline{\mu}_{\perp}}{\overline{\mu}_{\|}}\frac{\sin\theta-\overline{\raisebox{2.0pt}{$\chi$}}}{\cos\theta}}\right).$
(46)

The deflection angle, defined as the difference between the phase (launching)
direction and the power direction, is then obtained as
$\theta_{\mathrm{defl}}=\theta_{\mathbf{S}}-\theta_{\mathbf{k}}$, where
$\theta_{\mathbf{k}}=\arctan(k_{z}/k_{x})$ is the phase angle. Using the
identity $\tan^{-1}(x)-\tan^{-1}(y)=\tan^{-1}((x-y)/(1+xy))$, this angle may
be expressed as

$\theta_{\mathrm{defl}}=\tan^{-1}\left(\frac{(\mu_{\perp}-\mu_{\|})k_{x}k_{z}-\mu_{\perp}k_{x}\omega\overline{\raisebox{2.0pt}{$\chi$}}/c}{\mu_{\perp}k_{x}^{2}+\mu_{\perp}k_{z}^{2}-k_{z}\omega\overline{\raisebox{2.0pt}{$\chi$}}/c}\right).$
(47)

###  C.3 Fields for Gaussian Beam Excitation

We consider the propagation of an s polarized plane-wave Gaussian beam that is
launched in the $x$ direction. The corresponding field in the plane $x=0$
reads

$E_{y}(z,x=0)=E_{0}\exp\left(-\frac{z^{2}}{W_{0}^{2}}\right),$ (48)

where $E_{0}$ and $W_{0}$ are the amplitude and the waist of the beam. Taking
the $z$-spatial Fourier transform of (48) yields

$\begin{split}\tilde{E}_{y}(k_{z};x=0)&=\frac{1}{2\pi}\int_{-\infty}^{+\infty}E_{0}\exp\left(-\frac{z^{2}}{W_{0}^{2}}\right)\text{e}^{-\mathrm{i}k_{z}z}dz\\\
&=\frac{W_{0}E_{0}}{2\sqrt{\pi}}\exp\left(-\frac{k_{z}^{2}W_{0}^{2}}{4}\right).\end{split}$
(49)

The spectral field in any plane $x$ may be obtained by multiplying (49) with
the propagator of the problem [65], $\text{e}^{ik_{x}x}$, namely

$\tilde{E}_{y}(k_{z},x)=\text{e}^{\mathrm{i}k_{x}x}\tilde{E}_{y}(k_{z},x=0),$
(50)

where $k_{x}$ is related to $k_{z}$ by (34). The field in (49) becomes then

$\tilde{E}_{y}(k_{z},x)=\frac{W_{0}E_{0}}{2\sqrt{\pi}}\exp\left(-\frac{k_{z}^{2}W_{0}^{2}}{4}\right)\text{e}^{\mathrm{i}k_{x}x},$
(51)

to which we apply the inverse Fourier transform to provide the field at any
point $z$ and $x$,

$E_{y}(z,x)=\frac{W_{0}E_{0}}{2\sqrt{\pi}}\int_{-\infty}^{+\infty}\exp\left(-\frac{k_{z}^{2}W_{0}^{2}}{4}\right)\text{e}^{\mathrm{i}(k_{x}x+k_{z}z)}dk_{z}.$
(52)

Given the assumed $x$-direction of propagation and the Gaussian nature of
beam, $k_{z}\ll k_{0}$ ($k_{0}=\omega/c$), $k_{x}$ in (34) may be approximated
as

$k_{x}\approx\sqrt{\overline{\epsilon}_{\|}\overline{\mu}_{\perp}}k_{0}\left(1-\frac{(k_{z}-\overline{\raisebox{2.0pt}{$\chi$}}k_{0})^{2}}{2\overline{\epsilon}_{\|}\overline{\mu}_{\|}k_{0}^{2}}\right),$
(53)

which is a modulation-motion bianisotropic generalization of the paraxial
approximation. Inserting this expression into (52), rearranging and factoring
out the $k_{z}$-independent terms yields

$\begin{split}E_{y}(z,x)=&\frac{E_{0}W_{0}}{2\sqrt{\pi}}\text{e}^{\mathrm{i}\sqrt{\overline{\epsilon}_{\|}\overline{\mu}_{\perp}}k_{0}\left(1-\tfrac{\overline{\chi}^{2}k_{0}}{2\overline{\epsilon}_{\|}\overline{\mu}_{\|}}\right)x}\\\
&\quad\int_{-\infty}^{+\infty}\exp\left(-k_{z}^{2}\left(\frac{W_{0}^{2}}{4}+\mathrm{i}\frac{\sqrt{\overline{\epsilon}_{\|}\overline{\mu}_{\perp}}}{\overline{\epsilon}_{\|}\overline{\mu}_{\|}}\frac{x}{2k_{0}}\right)\right)\\\
&\quad\quad\text{e}^{\mathrm{i}\frac{\sqrt{\overline{\epsilon}_{\|}\overline{\mu}_{\perp}}}{\overline{\epsilon}_{\|}\overline{\mu}_{\|}}\overline{\raisebox{2.0pt}{$\chi$}}xk_{z}}\text{e}^{ik_{z}z}dk_{z}.\end{split}$
(54)

Calculating the integral in (54) yields the final result,

$\begin{split}E_{y}(z,x)=&E_{0}\frac{\text{e}^{\mathrm{i}\sqrt{\overline{\epsilon}_{\|}\overline{\mu}_{\perp}}k_{0}\left(1-\tfrac{\overline{\chi}^{2}k_{0}}{2\overline{\epsilon}_{\|}\overline{\mu}_{\|}}\right)x}}{1+\mathrm{i}\frac{\sqrt{\overline{\epsilon}_{\|}\overline{\mu}_{\perp}}}{\overline{\epsilon}_{\|}\overline{\mu}_{\|}}\frac{2x}{k_{0}W_{0}^{2}}}\\\
&\quad\exp\left(-\frac{\left(z+\frac{\sqrt{\overline{\epsilon}_{\|}\overline{\mu}_{\perp}}}{\overline{\epsilon}_{\|}\overline{\mu}_{\|}}\overline{\raisebox{2.0pt}{$\chi$}}x\right)^{2}}{\left(1+\mathrm{i}\frac{\sqrt{\overline{\epsilon}_{\|}\overline{\mu}_{\perp}}}{\overline{\epsilon}_{\|}\overline{\mu}_{\|}}\frac{2x}{k_{0}W_{0}^{2}}\right)W_{0}^{2}}\right),\end{split}$
(55)

which features an $x$-dependent $z$ lateral shift.

Note that in the absence of modulation ($\beta=0$ and two identical layers),
the medium reduces to a simple _isotropic stationary medium_ , with (55)
correspondingly reducing to

$\begin{multlined}E_{y}(z,x)=E_{0}\frac{\text{e}^{\mathrm{i}kx}}{1+\mathrm{i}\frac{2x}{kW_{0}^{2}}}\exp\left(-\frac{z^{2}}{\left(1+\mathrm{i}\frac{2x}{kW_{0}^{2}}\right)W_{0}^{2}}\right),\end{multlined}E_{y}(z,x)=E_{0}\frac{\text{e}^{\mathrm{i}kx}}{1+\mathrm{i}\frac{2x}{kW_{0}^{2}}}\exp\left(-\frac{z^{2}}{\left(1+\mathrm{i}\frac{2x}{kW_{0}^{2}}\right)W_{0}^{2}}\right),$
(56)

where $k=nk_{0}$, where $n$ is the refractive index of the medium [65].

###  C.4 Numerical Validation

Figure 7 shows the FDTD simulation corresponding to the analytical result in
Fig. 3.

Figure 7: FDTD simulation corresponding to the analytical result in Fig. 3,
with FDTD mesh size $\Delta z=\Delta x=(3/2)10^{-5}$ m, time step $\Delta
t=(2/3)10^{-13}$ s, Mur’s absorbing boundary conditions and a computational
domain of area $417\Delta z\times 250\Delta x$ and duration $33000\Delta t$.

Figure 8 plots the logarithm of the difference between the analytical field
plotted in Fig. 3 and the FDTD field plotted in Fig. 7, with this difference
being as defined as

$\delta_{E_{y}}=\frac{\left|E_{y}^{\mathrm{analytical}}-E_{y}^{\mathrm{FDTD}}\right|}{E_{0}}.$
(57)

Figure 8: Difference $\log_{10}(\delta_{E_{y}})$, with range limited to the
interval $[0,-5]$ dB, corresponding to $\max\left(\delta_{E_{y}}\right)=5\%$,
between the analytical field plotted in Fig. 3 and the FDTD field plotted in
Fig. 7 .

We verified that the difference in Fig. 8 decreases with increasing FDTD mesh
density. This convergence of the FDTD simulation to the analytical result
validates the latter.

##  Appendix D Modulation Parameters for Black and White Holes

The electromagnetic parameters corresponding to a given gravitational metric
$g_{\mu\nu}$ are [66]

$\epsilon^{ij}=\mu^{ij}=-\frac{\sqrt{-g}}{g_{00}}g^{ij}$ (58a) for the
permittivity and permeability, and
$\raisebox{2.0pt}{$\chi$}_{j}=\frac{g_{0j}}{g_{00}}$ (58b)

for the chirality.

The metric of black or white holes is found from the Schwarzschild space-time
interval

$ds^{2}=-\left(1-\frac{2MG}{r}\right)(cdt)^{2}+\left(1-\frac{2MG}{r}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},$
(59a) where $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2},$ (59b)

as

$g_{00}=-\left(1-\frac{2MG}{r}\right)$ (60a) and
$g_{11}=\left(1-\frac{2MG}{r}\right)^{-1},\quad g_{22}=r^{2},\quad
g_{33}=r^{2}\sin^{2}\theta.$ (60b)

The Schwarzschild metric (60) has two singularities, one at the origin, $r=0$,
and one at the “Schwarzschild radius”, $r=2MG$. The former is an essential
singularity [67], whereas the latter is an artifact of the coordinates system
in (59) 666The metric associated with (59) is not conformally flat, i.e., its
manifold cannot be mapped to a flat space via a conformal transformation map
[50] and can therefore be removed with a proper change of coordinates [69]. An
appropriate coordinate system to remove the spatial singularity would be one
with the radial coordinate being transformed as [70]

$r=\hat{r}\left(1+\frac{2MG}{4\hat{r}}\right),$ (61)

where $\hat{r}$ is the new radial coordinate. Inserting (61) into (59a) leads
to the new space-time interval

$ds^{2}=-\left(\frac{1-\frac{2MG}{4\hat{r}}}{1+\frac{2MG}{4\hat{r}}}\right)^{2}(cdt)^{2}+\left(1+\frac{2MG}{4\hat{r}}\right)^{4}(d\hat{r}^{2}+\hat{r}^{2}d\Omega^{2}),$
(62a)

with its corresponding metric components

$g_{00}=-\left(\frac{1-\frac{2MG}{4\hat{r}}}{1+\frac{2MG}{4\hat{r}}}\right)^{2}$
(62b)

and

$\begin{split}&g_{11}=\left(1+\frac{2MG}{4\hat{r}}\right)^{4},\quad
g_{22}=\left(1+\frac{2MG}{4\hat{r}}\right)^{4}r^{2},\quad\\\
&g_{33}=\left(1+\frac{2MG}{4\hat{r}}\right)^{4}r^{2}\sin^{2}\theta.\end{split}$
(62c)

and the equivalent medium parameters obtained by inserting (62b) and (62c)
into (58)

$\epsilon_{\mathrm{H}}=\mu_{\mathrm{H}}=\frac{(1+\frac{2MG}{4\hat{r}})^{3}}{1-\frac{2MG}{4\hat{r}}},$
(63a) $\raisebox{2.0pt}{$\chi$}_{\mathrm{H}}=0.$ (63b)

For 2D, we transform the coordinates from spherical ($\hat{r},\theta,\phi$) to
polar ($\rho,\phi$). Now given the circular symmetry of the medium parameter
in the new coordinates, we choose a circularly-symmetric modulation.
Therefore, the modulation has to move in radial direction ($\rho$).

For modulation with radial velocity ($\beta=\beta_{\rho}$) and radial
acceleration ($a^{\prime}=a^{\prime}_{\rho}$) in the $z-x$ plane the averaged
constitutive parameters can be derived in the same manner as (35a) to (35e) by
applying the continuity on the field components of $E_{y}$, $E_{\phi}$,
$H_{y}$, $H_{\phi}$, $D_{r}$ and $B_{r}$, namely

$\overline{\epsilon}_{y}=\overline{\epsilon}_{\phi}=\overline{\epsilon}_{\|}=\frac{\Sigma_{\epsilon}-\beta^{2}\epsilon_{1}\epsilon_{2}\Sigma_{\mu}}{1-\beta^{2}\Sigma_{\epsilon}\Sigma_{\mu}},$
(64a)
$\overline{\epsilon}_{\rho}=\overline{\epsilon}_{\perp}=\frac{2\epsilon_{1}\epsilon_{2}}{\Sigma_{\epsilon}},$
(64b)
$\overline{\mu}_{y}=\overline{\mu}_{\phi}=\overline{\mu}_{\|}=\frac{\Sigma_{\mu}-\beta^{2}\mu_{1}\mu_{2}\Sigma_{\epsilon}}{1-\beta^{2}\Sigma_{\epsilon}\Sigma_{\mu}},$
(64c)
$\overline{\mu}_{\rho}=\overline{\mu}_{\perp}=\frac{2\mu_{1}\mu_{2}}{\Sigma_{\mu}}$
(64d) and
$\overline{\raisebox{2.0pt}{$\chi$}}_{\rho}=\overline{\raisebox{2.0pt}{$\chi$}}=\beta\frac{\Delta_{\epsilon}\Delta_{\mu}}{1-\beta^{2}\Sigma_{\epsilon}\Sigma_{\mu}},$
(64e)

where $\Sigma_{\epsilon}=(\epsilon_{1}+\epsilon_{2})/2$,
$\Delta_{\epsilon}=(\epsilon_{1}-\epsilon_{2})/2$ and similarly for
$\Sigma_{\mu}$, $\Delta_{\mu}$.

Due to the time-reversal symmetry of the metric in (62a) ($g_{0j}=0$) the
equivalent bianisotropy term in (58b)($\raisebox{2.0pt}{$\chi$}_{j}$) is zero.
To have zero bianisotropy we need to have either the permittivity or the
permeability modulated. Given the assumption of s polarization (electric field
only along $y$) and the isotropic nature of the metric in the polar
coordinates we only modulate the permittivity, and leave the permeability
unmodulated ($\overline{\mu}=\mu_{1}=\mu_{2}$) Now we find the $a^{\prime}$,
$\beta_{0}$ and $\epsilon_{1,2}$ that satisfy the following equation

$\sqrt{\overline{\epsilon}_{\|}\overline{\mu}}=\sqrt{\epsilon_{\mathrm{H}}\mu_{\mathrm{H}}}.$
(65)

This procedure is carried out numerically and the FDTD simulation results of
such equivalent modulation parameters are shown in Fig. 5.

##  Appendix E Light Trajectories in Schwarzschild Gravity Analogs

Figure 9 depicts the qualitative trajectories of light near Schwarzschild
gravity analogs, with Fig. 9(a) corresponding to accelerated matter and Fig.
9(b) corresponding to accelerated modulation.

Figure 9: Qualitative light trajectories for Schwarzschild gravity analogs
based on (a) matter [71] and (b) modulation, with centripetal acceleration,
corresponding to a black holes (top panels), and centrifugal acceleration,
corresponding to a white holes (bottom panels).

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