figure table

# Photonics of Time-Varying Media

Emanuele Galiffi  Photonics Initiative, Advanced Science Research Center, City
University of New York, 85 St. Nicholas Terrace, 10031, New York, NY, USA  The
Blackett Laboratory, Department of Physics, Imperial College London, Prince
Consort Road SW2AZ, London, UK  Romain Tirole† The Blackett Laboratory,
Department of Physics, Imperial College London, Prince Consort Road SW2AZ,
London, UK  Shixiong Yin† Photonics Initiative, Advanced Science Research
Center, City University of New York, 85 St. Nicholas Terrace, 10031, New York,
NY, USA  Huanan Li  Photonics Initiative, Advanced Science Research Center,
City University of New York, 85 St. Nicholas Terrace, 10031, New York, NY, USA
School of Physics, Nankai University, Tianjin 300071, China  Stefano Vezzoli
The Blackett Laboratory, Department of Physics, Imperial College London,
Prince Consort Road SW2AZ, London, UK  Paloma A. Huidobro  Instituto de
Telecomunicações, Instituto Superior Tecnico-University of Lisbon, Avenida
Rovisco Pais 1, Lisboa, 1049-001 Portugal  Mário G. Silveirinha  Instituto de
Telecomunicações, Instituto Superior Tecnico-University of Lisbon, Avenida
Rovisco Pais 1, Lisboa, 1049-001 Portugal  Riccardo Sapienza  The Blackett
Laboratory, Department of Physics, Imperial College London, Prince Consort
Road SW2AZ, London, UK  Andrea Alù  Photonics Initiative, Advanced Science
Research Center, City University of New York, 85 St. Nicholas Terrace, 10031,
New York, NY, USA  Physics Program, Graduate Center, of the City University of
New York, New York, NY 10016, USA  J. B. Pendry  The Blackett Laboratory,
Department of Physics, Imperial College London, Prince Consort Road SW2AZ,
London, UK

###### Abstract

Time-varying media have recently emerged as a new paradigm for wave
manipulation, thanks to the synergy between the discovery of novel, highly
nonlinear materials, such as epsilon-near-zero materials, and the quest for
novel wave applications, such as magnet-free nonreciprocity, multi-mode light
shaping, and ultrafast switching. In this review we provide a comprehensive
discussion of the recent progress achieved with photonic metamaterials whose
properties stem from their modulation in time. We review the basic concepts
underpinning temporal switching and its relation with spatial scattering, and
deploy the resulting insight to review photonic time-crystals and their
emergent research avenues such as topological and non-Hermitian physics. We
then extend our discussion to account for spatiotemporal modulation and its
applications to nonreciprocity, synthetic motion, giant anisotropy,
amplification and other effects. Finally, we conclude with a review of the
most attractive experimental avenues recently demonstrated, and provide a few
perspectives on emerging trends for future implementations of time-modulation
in photonics.

###### keywords:

time-varying, temporal modulation, metamaterials, switching, optics,
photonics, light

*Emanuele Galiffi<EMAIL_ADDRESS>
$\dagger$ These authors contributed equally.

###### Contents

  1. 1 Introduction
  2. 2 Wave engineering with temporal interfaces
    1. 2.1 Temporal boundary conditions and temporal scattering
    2. 2.2 Temporal slabs
    3. 2.3 Temporal switching in anisotropic media
    4. 2.4 Temporal switching in the presence of material dispersion
    5. 2.5 Time-interfaces in spatial structures
  3. 3 Photonic Time-Crystals
    1. 3.1 Topology in photonic time-crystals
    2. 3.2 Non-Hermiticity and disorder
    3. 3.3 Time-modulated surfaces
  4. 4 Spatiotemporal metamaterials
    1. 4.1 Compact nonreciprocal devices without magnetic bias
    2. 4.2 Synthetic motion with spacetime media
    3. 4.3 Luminal amplification and spatiotemporal localization
    4. 4.4 Space-time metasurfaces
    5. 4.5 Further directions
  5. 5 Experimental advances and challenges in all-optical implementations
    1. 5.1 Photocarrier excitation and nonlinear optical modulation
      1. 5.1.1 Epsilon-Near-Zero materials, transparent conducting oxides and nonlinear optics
      2. 5.1.2 Hybrid platforms for time-varying experiments
      3. 5.1.3 High-index dielectric metasurfaces as time-varying media
    2. 5.2 New leads for time-varying metasurfaces
      1. 5.2.1 Quantum well polaritons
      2. 5.2.2 Magneto-optical modulation
      3. 5.2.3 Multilayered metamaterials
      4. 5.2.4 2D materials
    3. 5.3 Outlook
  6. 6 Conclusions

##  1 Introduction

In the wake of the extraordinary scientific advances of the last two
centuries, the role of time underpins many of the unsolved puzzles of the
physical world. In quantum theory, time is the only quantity not associated to
an observable, being merely treated as a variable, while attempts to bring it
to a common ground with all other physical quantities are still in the making
[1]. Meanwhile, at cosmological scales, the measured acceleration of the
expansion of the universe makes it hard to imagine a complete cosmological
theory where temporal dynamics of e.g. wave phenomena does not play a central
role. Yet, most basic physics is carried out under the assumption that
physical systems are passive, merely responding “after” any input stimuli.

Occasionally, the need to account for certain truly multi-physics or nonlinear
phenomena brings up opportunities to develop our understanding of wave physics
in temporally inhomogeneous systems: for instance, in order to realize bench-
top analogues of relativistic phenomena early attempts were made to tap into
time-modulated classical wave systems [2, 3]. However, the critical mass of
research efforts needed to accomplish sizeable advances in this direction has,
until recently, been lacking. On yet another front of exploration, time-
crystals have been proposed in condensed matter theory for almost a decade as
a stable state of matter capable of spontaneously breaking continuous time-
translation symmetry while preserving long-range temporal order [4]: their
very existence has been questioned and revised [5], and several increasingly
successful implementations have recently been accomplished [6, 7].

Meanwhile, over the past two decades, the rise of the field of metamaterials
has proven how much fundamental wave physics lies untapped under the hood of
well-established classical theories such as electromagnetism, acoustics and
elasticity. Recently, the quest for novel forms of wave-matter interactions in
these fields has led to a significant growth of interest in the exploration of
time as a novel degree of freedom for metamaterials [8]. In this pursuit, our
need for even a basic understanding of wave phenomena in the time domain has
reflected in the wealth of surprising physical effects recently unveiled,
largely theoretically and in part experimentally, which can be enabled by
externally applying explicit temporal modulations on the parameters of a
physical system.

In this Review Article we present an overview of the state of the art across
the rapidly expanding field of time-varying photonic metamaterials, in the
hope that it may serve the photonics and metamaterials community as a catalyst
for current and future explorations of this fascinating field. Whilst not as
detailed as a tutorial paper, the review aims at presenting the basic building
blocks of time-varying electromagnetics, with the main goal of developing
basic insight into the relevant phenomenology, while offering a broad view of
key past and future research directions in the field.

The paper is structured as follows: Section 2 starts from the basics of
electromagnetic time-switching, expanding towards more complex scenarios such
as such as non-Hermitian and anisotropic switching, temporal slabs and space-
time interfaces. Section 3 is dedicated to photonic time-crystals, namely
electromagnetic systems undergoing infinite (periodic or disordered)
modulations of their parameters in time-only, and we discuss some of its most
promising directions towards e.g. the engineering of topological phases,
synthetic frequency dimensions, non-Hermitian physics and localization,
concluding with some perspective on the peculiarities of time-varying
surfaces. Spatiotemporal crystals are discussed in Sec. 4, where we introduce
the peculiarities of space-time band diagrams, homogenization procedures for
travelling-wave modulated systems enabling effective-medium descriptions of
synthetic motion, and highlight some recently emerged opportunities for
nonreciprocity, synthetic Fresnel drag and wave amplification and compression,
as well as space-time modulated metasurface and some of their applications.
Finally, Sec. 5 offers an overview of the most promising platforms for
experimental implementations of time-modulation in photonics.

##  2 Wave engineering with temporal interfaces

Figure 1: Scattering at (a) a spatial discontinuity, generating transmitted
and reflected waves propagating in different media with refractive indices
$n_{1}$ and $n_{2}$, and (b) a temporal discontinuity, generating forward and
backward waves propagating in the same medium. (c,d) Illustration of the
respective scattering processes on a dispersion diagram: the blue and red
lines denote the dispersion cone for the two media. Note how the frequency
$\omega$ is conserved in a spatial scattering process, and coupling occurs to
forward and backward modes in the two different media, whereas in a time-
scattering process the momentum $k$ is conserved, and the two scattered waves
are both embedded in the new medium.

Light scattering by spatial interfaces [Fig. 1(a)] is at the basis of many
wave phenomena, from the simplest refraction effects to the complexity of wave
engineering in complex media involving multilayers[9], anisotropy[10],
nonlinearity[11], extreme constitutive parameters[12, 13, 14], or
metamaterials and metasurfaces[15], to name a few. How do electromagnetic
waves behave when encountering interfaces in time rather than in space?
Temporal interfaces can be created by abruptly switching the material
properties of a medium in time, preserving its spatial continuity [Fig. 1(b)].
Wave propagation in switched media was firstly investigated by Morgenthaler in
1958[16]. In parallel, the study of light interactions with abruptly switched
materials was developed in the plasma physics community, since the plasma
permittivity can be switched in time by fast ionization processes[17]. In this
section, we focus on the recent explorations on time-switching in photonic
materials.

###  2.1 Temporal boundary conditions and temporal scattering

It is well-known that wave propagation across spatial interfaces is governed
by established boundary conditions, see e.g., Ref. 18. The essential
constraint is that all relevant physical quantities, for instance, electric
and magnetic fields ($\mathbf{E}$ and $\mathbf{H}$), must be finite at all
points in space and time. Spatial boundary conditions are therefore derived by
integrating Maxwell’s equations over an infinitesimal area and volume across
the spatial interface, implying the continuity of tangential $\mathbf{E}$ and
$\mathbf{H}$ at an interface that does not support surface current densities.
Dually, the integration over a time interval across a temporal boundary obeys
temporal boundary conditions. Let us start with the macroscopic Maxwell’s curl
equations:

$\displaystyle\frac{\partial\mathbf{B}}{\partial t}$
$\displaystyle=-\nabla\times\mathbf{E},$ (1)
$\displaystyle\frac{\partial\mathbf{D}}{\partial t}$
$\displaystyle=-\mathbf{J}+\nabla\times\mathbf{H}.$ (2)

Here we assume that the medium is unbounded and homogeneous before and after a
switching instance at $t=t_{0}$. Integrating from $t_{0}^{-}=t_{0}-\epsilon$
to $t_{0}^{+}=t_{0}+\epsilon$ with a vanishing $\epsilon$, we expect that the
right-hand sides of Eqs. 1 and 2 are both zero, due to the finite values of
fields and sources. Then we obtain the temporal boundary conditions:

$\displaystyle\mathbf{B}\left(t=t_{0}^{+}\right)$
$\displaystyle=\mathbf{B}\left(t=t_{0}^{-}\right),$ (3a)
$\displaystyle\mathbf{D}\left(t=t_{0}^{+}\right)$
$\displaystyle=\mathbf{D}\left(t=t_{0}^{-}\right).$ (3b)

They ensure that the magnetic flux density $\mathbf{B}$ and the electric
displacement $\mathbf{D}$ vary continuously in the time domain. Alternatively,
the continuity of $\mathbf{B}$ and $\mathbf{D}$ can also be interpreted by the
conservation of magnetic flux $\Phi$ and electric charges $Q$, as shown by
Morgenthaler[16] and Auld[19], respectively.

One of the most intriguing phenomena that the temporal discontinuity in a
medium brings about is that it can scatter waves in a way dual to spatial
interfaces, producing forward and backward waves in time, rather than waves
reflected and transmitted in space[16, 19, 20, 21, 22, 23]. The scattering
coefficients can be found by applying the above temporal boundary conditions.
Consider a monochromatic plane wave traveling in an unbounded, homogeneous,
isotropic, non-dispersive but time-varying medium. Both permittivity
$\varepsilon$ and permeability $\mu$ are assumed to abruptly switch from their
initial values $\varepsilon_{1}$ and $\mu_{1}$ to $\varepsilon_{2}$ and
$\mu_{2}$ at $t=t_{0}$. We can denote the incident wave with
$D_{x}=D_{1}e^{j\omega_{1}t-jkz}$ and $B_{y}=B_{1}e^{j\omega_{1}t-jkz}$, where
$\omega_{1}$ is the angular frequency and $k$ is the wavenumber. The field
amplitudes are related by $B_{1}=Z_{1}D_{1}$ with
$Z_{1}=\sqrt{\mu_{1}/\varepsilon_{1}}$. The fields after the switching event
read

$\displaystyle D_{x}\left(t>t_{0}\right)$
$\displaystyle=\left[Te^{j\omega_{2}\left(t-t_{0}\right)}+Re^{-j\omega_{2}\left(t-t_{0}\right)}\right]D_{1}e^{j\left(\omega_{1}t_{0}-k_{2}z\right)},$
(4a) $\displaystyle B_{y}\left(t>t_{0}\right)$
$\displaystyle=Z_{2}\left[Te^{j\omega_{2}\left(t-t_{0}\right)}-Re^{-j\omega_{2}\left(t-t_{0}\right)}\right]D_{1}e^{j\left(\omega_{1}t_{0}-k_{2}z\right)}.$
(4b)

$T$ and $R$ are the transmission and reflection coefficients defined for the
electric displacement field, and $Z_{2}=\sqrt{\mu_{2}/\varepsilon_{2}}$ is the
new wave impedance after switching. The temporal boundary conditions (Eq. 3)
need to be satisfied everywhere in space, which implies that $k_{2}=k$, or
equivalently

$\displaystyle\omega_{2}\sqrt{\varepsilon_{2}\mu_{2}}$
$\displaystyle=\omega_{1}\sqrt{\varepsilon_{1}\mu_{1}}.$ (5)

This condition ensures momentum conservation across temporal interfaces, which
is expected due to the preserved spatial homogeneity[16, 17, 24, 25]. By
equating the incident and scattered $\mathbf{D}$ and $\mathbf{B}$ fields at
$t=t_{0}$, we solve for the temporal scattering coefficients:

$\displaystyle T$ $\displaystyle=\frac{Z_{2}+Z_{1}}{2Z_{2}},$ (6a)
$\displaystyle R$ $\displaystyle=\frac{Z_{2}-Z_{1}}{2Z_{2}}.$ (6b)

The coefficients for the electric field can be easily found by substituting
the constitutive relation between $D$ and $E$, which are adopted more widely
in the literature. It should be emphasized that the results presented in Eqs.
5 and 6 generally apply to an abrupt temporal interface under the assumptions
of spatial homogeneity, isotropy and instantaneous response of materials, and
may change if one or more of these assumptions are relaxed[26]. Illustrations
of spatial and temporal scattering processes are given in Fig. 1(a,c) and
(b,d) respectively. As we consider more realistic and complicated
electrodynamic models of materials, such as anisotropy, dispersion, and broken
translational symmetry in space, the temporal scattering is expectedly
modified[27, 28, 29, 30, 31, 32, 33]. To date, research on these topics is
very active, as discussed in the following subsections.

###  2.2 Temporal slabs

Leveraging scattering phenomena at time interfaces, research efforts have been
dedicated to the engineering of wave interference in time, for example in the
context of discrete time crystals[34, 35] and time metamaterials[36]. Even
when multiple temporal interfaces are involved, temporal interferences occur
only between forward and backward waves induced by each boundary
independently, due to the irreversibility of time, which introduces
significant differences compared to spatial interfaces. A key contrast between
spatial and temporal scattering from a mathematical standpoint is that space-
scattering generally gives rise to boundary-value problems, whereas time-
scattering leads to initial-value problems. Here we discuss the recent
progress in temporal slabs comprising one or two consecutive switching events.
In Sec. 3, we discuss (quasi-)periodically switched media.

The temporal interference introduced by a single temporal slab was firstly
investigated by Mendonça and Martins in Ref. 37, where they switched the
refractive index of a medium from $n_{0}$ to $n_{1}$, and then switched it
back to $n_{0}$ after a time interval $\tau$. The total transmission and
reflection amplitudes were found to be

$\displaystyle T_{total}$
$\displaystyle=\left[\cos\left(\omega_{1}\tau\right)+\frac{j}{2\alpha}\left(1+\alpha^{2}\right)\sin\left(\omega_{1}\tau\right)\right]e^{-j\omega_{0}\tau},$
(7a) $\displaystyle R_{total}$
$\displaystyle=-\frac{j}{2\alpha}\left(1+\alpha^{2}\right)\sin\left(\omega_{1}\tau\right)e^{j\omega_{0}\tau},$
(7b)

where $\alpha=n_{0}/n_{1}$ and $\omega_{0}$, $\omega_{1}$ are the frequencies
of waves outside and inside of the temporal slab, respectively. Similar
results were also obtained using a quantum optics formulation, illustrating
the probability of creating Fock states of photon pairs with opposite momentum
from the vacuum state[37]. A temporal Fabry-Perot slab with dispersion was
recently studied in Ref. 33, with slightly different results due to
dispersion. The most important phenomenon induced in these systems may
arguably be the amplification of light. Because frequency and energy are not
conserved at time boundaries, we can indeed expect large amplification of the
input energy at a suitably tailored temporal slab. It can be seen from Eq. 7
that such amplification process attains the maximum level when time interval
$\omega_{1}\tau$ is an odd multiple of $\pi/2$. On the contrary, time
reflections can be minimized with a different choice of switching intervals.
This finding can be connected to the dual phenomenon of impedance-matching
layers in spatial scattering. Indeed, Pacheco-Peña and Engheta have recently
introduced antireflection temporal coatings by rapidly changing the
permittivity of an unbounded medium twice[38], making an analogy to the
quarter-wavelength impedance matching in the spatial domain. The schematic of
such coatings is shown in Fig. 2(a). To minimize the backward reflection when
the permittivity is to be switched from $\varepsilon_{1}$ to
$\varepsilon_{2}$, they introduced a temporal coating with permittivity
$\varepsilon_{eq}$ and duration $\Delta t$ in between. Impedance matching is
enabled with

$\displaystyle\varepsilon_{eq}$
$\displaystyle=\sqrt{\varepsilon_{1}\varepsilon_{2}},\Delta
t=\frac{nT_{eq}}{4};n=1,3,5,...$ (8)

where $T_{eq}=1/f_{eq}$ is the equivalent period of the wave right after the
permittivity is switched to $\varepsilon_{eq}$, and the equivalent frequency
$f_{eq}=\sqrt{\varepsilon_{1}/\varepsilon_{eq}}f_{1}$. A numerical simulation
is shown in Fig. 2(b), indicating nearly vanishing reflection from the
temporal interfaces. Although Eq. 8 looks similar to the condition for
impedance matching in the spatial case[39], it is more than a dual to its
spatial counterpart. Firstly, the frequency is shifted from $f_{1}$ to
$f_{2}=\sqrt{\varepsilon_{1}/\varepsilon_{2}}f_{1}$, as shown in Fig. 2(c). In
addition, the energy and power flow of the waves have also been modified as we
change the permittivity of the material.

More recently, it has been shown that extreme power and energy manipulation
can be achieved in these types of temporal slabs once loss and gain are
considered. In analogy to conventional parity-time (PT) symmetry with balanced
gain and loss[40], Li et al. investigated a scenario where a pair of temporal
slabs obey temporal parity-time (TPT) symmetry[41]. In their work, the
temporal parity operation flips the arrow of time, while the “time” operation
is defined to reverse the waves in space. TPT-symmetric temporal slabs are
therefore realized if an unbounded medium is switched from a positive to a
negative conductivity in time, $\sigma_{2}=-\sigma_{3}>0$, for equal duration
$\Delta t$ before switching back to the initial Hermitian medium, as shown in
Fig. 2(d). Instantaneous material responses for both permittivity and
conductivity have still been assumed here.

Figure 2: (a) Schematic of antireflection temporal coatings proposed in Ref.
38. (b) Numerical results for the field distribution before and after the
temporal scattering, showing the incident and transmitted waves, with
minimized reflection due to the temporal coating shown in (a). (c) Spectra of
the incident, forward and backward waves. (d) Schematic of temporal parity-
time (TPT) symmetric structures proposed in Ref. 41. (e) and (f): The time
evolution of the normalized energy (solid curves) and its two constitutions
(circle symbols for interferences and the dashed curves for another). (e)
depicts the case of maximum total power while (f) shows that of minimum total
power. Insets are the field distributions in simulations before, during and
after the TPT slabs. Figures adapted from Refs. 38, 41.

By introducing a general temporal scattering matrix formulation, which may be
powerfully applied to any multi-layer configuration of temporal interfaces,
such (temporally finite) TPT-symmetric structures have been proven to always
reside in their symmetric phase, for which the scattering matrix supports
unimodular eigenvalues. Interestingly, the non-orthogonal nature of wave
interference in non-Hermitian media enables exotic energy exchanges, despite
the fact that the overall temporal bilayer is in its symmetric phase. As a
result, the total power flow carried by the waves after the switching events,
$P_{tot}(t)=P^{+}(t)+P^{-}(t)$ with $P^{\pm}(t)$ for the forward/backward
propagating wave, can be significantly different from the incident one. By
controlling the relative phase of two incident counter-propagating waves, we
can reach drastically different total power flows for the same TPT-symmetric
temporal bilayer. As an example, for $n_{1}=1$, $n_{2}=2-j0.2$ and
$2\omega_{1}\Delta t=3.5$, the temporal evolution of the normalized stored
energy density (solid curves) and its two components
$U_{i}=2\varepsilon\left[|E^{+}(t)|^{2}+|E^{-}(t)|^{2}\right]$ (dashed curves,
proportional to the total power flow in the two waves $E^{+}(t)$ and
$E^{-}(t)$, and equal to the total stored energy in Hermitian media) and
$U_{c}=2\mathrm{Re}\left[n(n^{*}-n)E^{+}(t)E^{-}(t)^{*}\right]$ (circle
symbols, stemming from the non-orthogonality of the counter-propagating waves
in non-Hermitian media) are shown in Fig. 2(e,f) for two different relative
phases of the input waves, yielding widely different power flows after the
switching events. Thus, dramatic power amplification and attenuation can be
achieved in such non-Hermitian temporal bilayers as a function of the relative
phase of the excitations, as a dual phenomenon to laser-absorber pairs in
conventional PT-symmetric systems.

###  2.3 Temporal switching in anisotropic media

Recent research has added new degrees of freedom to enable exotic wave
transformations based on time-switching. One intriguing possibility is to
consider material anisotropy and spatial dispersion. In 2018, Akbarzadeh et
al. raised the question of whether it is possible to realize an analogue of
Newton’s prism, illustrated in Fig. 3(a)[28], in a time metamaterial. To map
spatial frequencies into temporal frequencies, not only temporal invariance
needs to be violated, but also spatial symmetry breaking is required to bridge
different momenta with different frequency channels, as shown in Fig. 3(b).
The result is an inverse prism, as presented in Fig. 3(c), in which an
unbounded isotropic medium with scalar refractive index $n_{1}$ is switched to
a uniaxial medium with
$\bar{\bar{n}}_{2}=\mathrm{diag}\left(n_{\parallel},n_{\parallel},n_{\perp}\right)$
at $t=t_{0}$. The right part of panel (c) represents the isofrequency curves
of the medium before and after switching. Because the medium remains
homogeneous, the conservation of wavenumber $\mathbf{k}$ (momentum) still
holds, leading to “birefringence” in temporal frequencies. A monochromatic
wave at frequency $\omega_{1}$ propagating in the isotropic medium before
switching will then be mapped to different frequencies, depending on its
polarization and momentum, following

$\displaystyle n_{1}\omega_{1}=\pm n\left(\mathbf{k}\right)=\begin{cases}\pm
n_{\parallel}\omega_{2}&s\text{-polarization }\\\ \frac{\pm
n_{\parallel}n_{\perp}\omega_{2}}{\sqrt{n_{\parallel}^{2}\cos^{2}\theta}+n_{\perp}^{2}\sin^{2}\theta}&p\text{-polarization
}\end{cases}$ (9)

with $\theta=\tan^{-1}\left(k_{z}/k_{x}\right)$ as defined in Fig. 3(c). In
general, linearly polarized light experiencing such an inverse prism will
feature Lissajous polarization with a time-varying phase difference between
two orthogonal field components, due to the distinct values of $\omega_{2}$
for s and p-polarizations.

One interesting phenomenon for wave propagation in anisotropic crystals is
that the group velocity does not necessarily align with the phase
velocity[10]. Based on this feature, the concept of temporal aiming was
proposed in Ref. 42. Figure 3(d) provides an overview of this phenomenon based
on isotropic-to-anisotropic switching. A p-polarized wave packet is immersed
in a time-switched medium. Different switching schemes
$\bar{\bar{\varepsilon}}_{r}(t)$ shown in the insets would route the signal to
different receivers (denoted Rx 1-3 in Fig. 3(d)). For an incoming wave with
propagation angle $\theta_{1}=\tan^{-1}\left(k_{z}/k_{x}\right)$, the
direction of momentum $\mathbf{k}$ after switching to the anisotropic medium
remains the same $\theta_{2k}=\theta_{1k}=\theta_{1}$, while that of the
Poynting vector $\mathbf{S}$ is redirected to

$\displaystyle\theta_{2S}$
$\displaystyle=\tan^{-1}\left(\frac{\varepsilon_{r2z}}{\varepsilon_{r2x}}\tan\theta_{1}\right),$
(10)

Figure 3: (a) A conventional prism decomposing white light into its frequency
components in different directions. (b) An inverse prism that maps the light
with different momentum into different frequencies. (c) Implementation of the
inverse prism proposed in Ref. 28. (d) Temporal aiming proposed in Ref. 42.
(e) The conventional Brewster angle. (f) Temporal Brewster angle described in
Ref. 43. Figures adapted from Refs. 28, 42, 43.

assuming that the anisotropic medium is a uniaxial crystal with
$\bar{\bar{\varepsilon}}_{2r}=\mathrm{diag}\left(\varepsilon_{2x},\varepsilon_{2x},\varepsilon_{2z}\right)$.
The discrepancy between $\theta_{2S}$ and $\theta_{1}$ plays a key role in the
aiming process by enforcing that the wave packet drifts transversely to the
initial propagation direction. After an appropriate duration, the medium is
then switched back to the initial state to allow the signal to travel again at
the same frequency and direction. The idea of signal aiming through temporal
switching paves the way to routing waves through temporal interfaces, creating
a form of temporal waveguiding.

Another interesting phenomenon exploiting temporal switching with anisotropic
media is the temporal Brewster angle[43]. The conventional Brewster angle is
defined as the angle at which the reflection of p-polarized incidence
vanishes, as shown in Fig. 3(e). Its temporal counterpart is illustrated in
Fig. 3(f), and similarly determined by the condition that no backward wave is
produced at the time interface. The temporal Brewster angle is given by the
following simple expression:

$\displaystyle\theta_{tB}$
$\displaystyle=\sin^{-1}\sqrt{\frac{\varepsilon_{r2x}\left(\varepsilon_{r2z}-\varepsilon_{r1}\right)}{\varepsilon_{r1}\left(\varepsilon_{r2z}-\varepsilon_{r2x}\right)}}.$
(11)

Notice that the Brewster angle for s-polarized waves is also expected if we
consider biaxial anisotropy where
$\varepsilon_{2x}\neq\varepsilon_{2y}\neq\varepsilon_{2z}$. These results open
the new avenues in controlling the polarization of waves by exploiting
temporal interfaces. For instance, Xu et al. have reported complete
polarization conversion using anisotropic temporal slabs[44].

###  2.4 Temporal switching in the presence of material dispersion

All previous results have assumed that the materials involved are non-
dispersive, such that they respond to abrupt switching events instantaneously.
However, caution is needed about this assumption, as the material response may
have temporal dynamics comparable with the finite switching times of realistic
modulation processes. In general, the temporal nonlocal response (dispersion)
of a material can be considered by assuming a susceptibility kernel in the
form
$\mathbf{D}(t)=\varepsilon_{0}\left[\varepsilon_{\infty}\mathbf{E}(t)+\int
dt^{\prime}\chi(t,t^{\prime})\mathbf{E}(t-t^{\prime})\right]$, where
$\varepsilon_{\infty}$ is the background relative permittivity and
$\chi(t,t^{\prime})$ is the time-dependent electric susceptibility, which must
satisfy causality. Recently, the generalization of the Kramers-Kronig
relations have been introduced for both adiabatic[45] and non-adiabatic[46]
time-varying susceptibility $\chi$. Research on temporal switching with
material dispersion dates back to Fante’s and Felson’s work in 1970s[20, 21].
In parallel, wave propagation phenomena such as self-phase modulation and
frequency conversion were studied in rapidly growing plasmas due to
ionization[47, 48, 49, 50]. Comprehensive reviews on time-varying plasmas can
be found in Refs. 17 and 27.

Once material dispersion is not negligible, it has been shown that the
electric field $\mathbf{E}$ becomes continuous at a time interface, giving
rise to additional temporal boundary conditions in addition to Eq. 3. The
temporal boundary conditions in this scenario have been derived in Ref. 30
from the perspective of Parseval’s theorem and in Ref. 31 based on the balance
of distributions. In the time domain, a Drude-Lorentz medium features a
second-order differential equation for the electric polarization density
$\mathbf{P}$:

$\displaystyle\frac{d^{2}\mathbf{P}}{dt^{2}}+\gamma\frac{d\mathbf{P}}{dt}+\omega_{0}^{2}\mathbf{P}=\varepsilon_{0}\omega_{p}^{2}\mathbf{E},$
(12)

where $\gamma$ is the collision damping rate, $\omega_{0}$ is the natural
frequency of the free electron gas, and
$\omega_{p}=\sqrt{Ne^{2}/\left(m_{e}\varepsilon_{0}\right)}$ with the
volumetric carrier density $N$, and electron’s mass $m_{e}$ and charge $e$. A
time-switched $N$ was considered in Refs. 29, 30 to effectively change
$\omega_{p}$ abruptly. The boundary conditions at the temporal interface turn
out to be the continuity of $\mathbf{B}$, $\mathbf{D}$, $\mathbf{E}$ (and
therefore $\mathbf{P}$) and $d\mathbf{P}/dt$. Meanwhile, the conservation of
momentum $\mathbf{k}$ in general gives rise to four solutions for the new
frequencies $\pm\omega_{1,2}$.

###  2.5 Time-interfaces in spatial structures

While drawing significant attention, the concept of time switching still bears
the question about accessibility to practical implementations. Until now
experimental work demonstrating time-reversal at a temporal interface has been
limited to water-wave phenomena by Fink’s group, whereby a water tank was
uniformly shaken by an impulse, leading to the refocusing of the waves back to
their emission point [51]. In addition to limitations in the achievable
modulation speed and strength, one chief difficulty also lies in altering a
medium in its entirety. Huge energy exchanges and stringent simultaneity of
switching are typically required. Instead, more and more efforts have been
geared towards switching materials only for spatially finite structures or in
lower dimensions. For instance, temporal reflections were studied in Ref. 52,
53, 54 for two-dimensional graphene plasmons, as shown in Fig. 4(a). In one-
dimensional transmission lines, broadband and efficient impedance matching can
be achieved by time-switching, as shown in Fig. 4(b) from Ref. 55. Since time-
switching relaxes the constraint of time-invariance, the proposed matching
schemes can go beyond the conventional Bode-Fano efficiency bounds.

Taking advantage of enhanced light-matter interactions in finite structures
which support resonances, we can extend temporal switching to other
applications. Switching resonant cavities enables possibilities for plasma
radiation[56] and photon-generation[57]. Similar ideas have also been applied
to induce static-to-dynamic field radiation in a switched dielectric brick
sandwiched in a waveguide[58]. Li and Alù explored a time-switched Dallenbach
screen, whose schematic is shown in Fig. 4(c), showing that time-switching can
extend absorption bandwidth by changing the permittivity of an absorber whilst
a broadband signal enters it [59]. On a related note, Mazor et al. unveiled
how an excitation can be unitarily transferred between coupled cavities by
switching the coupling strength, even if the detuning between the cavities is
large[60]. An example of this phenomenon in a coupled LC resonator pair is
shown in Fig. 4(d): by properly switching the value of the coupling
admittance, the energy stored in the capacitor $C_{1}$ can be transferred to
$C_{2}$ efficiently. Subwavelength time-modulated meta-atoms have also been
explored by Tretyakov’s and Fleury’s groups [61], developing a consistent
description of the interplay between time-modulation and dispersive
polarizability [62], while Engheta’s group recently extended this concept to
anisotropic meta-atoms, using it to demonstrate temporal deflection[63], as
shown in Fig. 4(e).

Figure 4: (a) Temporal reflection of graphene plasmons reported in Ref. 53.
(b) Impedance matching using a time-switched transmission line [55]. (c) Time-
switched thin absorber in Ref. 59. (d) Unitary excitation transfer between two
coupled circuit resonators [60]. (e) Temporal deflection caused by isotropic-
to-anisotropic switching on meta-atoms [63]. (f) Temporal switching of
structural dispersion for ultrafast frequency-shifts, proposed in Ref. 64. (g)
Corresponding frequency spectra for the switching in panel (f). Figures
adapted from Refs. 53, 55, 59, 60, 63, 64.

Finally, abrupt changes to the structural dispersion of a cavity, realized
e.g. by switching part of a structure, constitute yet another intriguing
degree of freedom to be exploited. For instance, in Ref. 64, Miyamaru et al.
achieved ultrafast terahertz frequency-shifts by metalizing the upper boundary
of a single-metalized waveguide at sub-picosecond timescales via photocarrier
excitation, thereby modifying its dispersion relation and inducing a large
frequency shift, as shown in Fig. 4(f,g). Similarly, nonlinear terahertz
generation was reported later in ultrafast time-varying metasurfaces [65].
Abruptly altering a spatial boundary can be interpreted heuristically as
switching the effective permittivity of the waveguide structure[66], which
also enforces a temporal interface on the fields inside the waveguide.

To summarize, in this section we discussed wave manipulation mechanisms in the
presence of temporal interfaces. We revisited the temporal boundary conditions
for Maxwell’s Equations and highlighted a few representative works on temporal
scattering and interferences. From the theory side the field is open to the
study of more sophisticated material models, such as non-Hermitian,
anisotropic and even bianisotropic switching mechanisms, also in spatially
structured systems. In the quest for extreme wave phenomena, temporal
switching with exotic media has opened a brand-new avenue for electromagnetics
research and beyond, although its ultimate impact will hinge upon further
implementations of temporal switching experiments, some of which we will
discuss in Sec. 5.

##  3 Photonic Time-Crystals

Having discussed the arsenal of potential wave phenomena based on individual
time-switching, we now move on to consider systems whose properties are
periodically (or quasi-periodically) modulated in time. The simplest
mechanical archetype of such a parametric system is a pendulum, whose length
is periodically modulated in time, one historic example being the
“Botafumeiro”, an 80 kg incensory at the Cathedral of Santiago de Compostela,
whose chain length is periodically modulated by monks, reaching speeds of 68
km/h over arc lengths of 65-meter within 17 modulation cycles [67] (a more
familiar one is the child swing, whereby the height of the center of mass is
modulated). In fact, a characteristic feature of parametric amplification is
its exponential growth rate, as opposed to the linear growth that occurs in a
driven oscillator.

We can make use of the scattering coefficients derived in Sec. 2, to construct
the building blocks of parametric amplification in electromagnetism. After a
first time-switching leading to a change of impedance
$Z_{1}=\sqrt{\mu_{1}/\varepsilon_{1}}\to
Z_{2}=\sqrt{\mu_{2}/\varepsilon_{2}}$, the displacement field at time $t$ will
comprise forward and backward waves according to:
$D_{2}(t)\propto[T_{12}e^{j\omega_{2}t}+R_{12}e^{-j\omega_{1}t}]$. If we now
switch the material again after a time $\Delta t_{2}$, the new forward waves
will encompass doubly-reversed and doubly-transmitted waves. Conversely, the
backward waves will consist of waves which were reflected on only one of the
two switching events, giving, after an additional time $\Delta t_{3}$:

$\displaystyle D_{2}/D_{0}$ $\displaystyle\propto(T_{12}e^{j\omega_{2}\Delta
t_{2}}T_{23}+R_{12}e^{-j\omega_{2}\Delta t_{2}}R_{23})e^{j\omega_{3}\Delta
t_{3}}$ (13) $\displaystyle+(T_{12}e^{j\omega_{2}\Delta
t_{2}}R_{23}+R_{12}e^{-j\omega_{2}\Delta t_{2}}T_{23})e^{-j\omega_{3}\Delta
t_{3}}$

This argument can be easily extended for a given number of switching events,
with forward and backward waves consisting of scattering contributions
involving an even and odd number of time-reversal processes $R_{i,i+1}$
respectively.

In order to study in more detail the effect of a periodic modulation, let us
set the final parameters equal to the initial ones,
$\varepsilon_{3}=\varepsilon_{1}$ and $\mu_{3}=\mu_{1}$, which forms one
modulation cycle. Evaluating the energy content of the forward and backward
waves, we arrive at:

$\displaystyle|\mathbb{T}_{2}|^{2}$
$\displaystyle=1+\frac{1}{2}\frac{(Z_{1}^{2}-Z_{2}^{2})^{2}}{Z_{1}^{2}Z_{2}^{2}}\sin^{2}(\omega_{2}\Delta
t_{2})$ (14) $\displaystyle|\mathbb{R}_{2}|^{2}$
$\displaystyle=\frac{1}{2}\frac{(Z_{1}^{2}-Z_{2}^{2})^{2}}{Z_{1}^{2}Z_{2}^{2}}\sin^{2}(\omega_{2}\Delta
t_{2})$ (15)

so that it is evident that the relation:
$|\mathbb{T}_{2}|^{2}=1+|\mathbb{R}_{2}|^{2}$ must always hold true regardless
of the extent and duration of the “temporal slab”. In order to demonstrate the
generality of this argument for any number of switchings, it is instructive to
keep in mind here that momentum conservation must hold true across any number
of such scattering processes. Calculating either the Abraham (kinetic)
momentum $\mathbf{E}\times\mathbf{H}$ or the Minkowski (canonical) momentum
$\mathbf{D}\times\mathbf{B}$ must in fact yield the same total final momentum.
As a result, it should not come as a surprise that any amplification of
forward waves must be accompanied by an equal amplification (or generation) of
backward waves, so that the overall effect of the parametric pumping is to
effectively generate a standing wave on top of the incoming beam. Clearly, in
a realistic scenario, since the forward and backward waves are orthogonal in
the absence of modulation, they can be independently outcoupled, leading to a
net increase of energy in both forward and backward waves. As another
consequence of momentum conservation, the energy in the system cannot be
reduced: any change in the forward wave amplitude must be compensated by the
generation of a backward wave, which can only have a positive contribution to
the total energy in the system. Notice how this is dual to the spatial case,
where the total energy in the system must be conserved, so that the power in
the forward-scattered waves can only be reduced by a scattering process.
Finally, it is worth pointing out that that no energy exchange can occur from
time-switching if the system is impedance-matched. As we will discuss in Sec.
4, however, it is still possible to have gain in impedance-matched scenarios
within certain spatiotemporal modulation regimes [Sec. 4.3].

We now move on to study the key features of infinite time-periodic systems,
which we shall regard as “photonic time-crystals” (PTC), although it should be
noted that this term bears an ambiguity with the concept of time-crystal in
condensed matter physics [68], where it denotes a stable phase of matter which
spontaneously breaks continuous time-translation symmetry. Here we intend PTCs
as active systems relying on an externally induced time-periodic modulation of
their constitutive parameters.

In a conventional photonic space-crystal (PSC) it is well-known that Bragg
scattering between waves separated in momentum by integer multiples of the
reciprocal lattice vector $\mathbf{g}$ of the crystal leads to the formation
of band-gaps in energy, in correspondence with the high-symmetry points in
reciprocal space (in 1D, this would happen at $k=\pm g/2$). In these gaps, the
states have an imaginary momentum component, thus decaying into the crystal
(growing states are forbidden by energy conservation). It is expected,
therefore, that modulating a system periodically in time at frequency $\Omega$
will yield band gaps in momentum ($k$-gaps) near the frequency $\Omega/2$,
where $\Omega$ plays the role of a reciprocal lattice vector, as a result of
the interference between the waves which are forward scattered by the
modulation, and those which are time-reversed. However, while in a PSC the
wavevector is imaginary inside the band-gap, in a PTC it is the energy that
acquires an imaginary component within a k-gap. As expected from our
discussion on momentum and energy conservation in the previous section, while
waves must spatially decay into the PSC to preserve energy, they can only grow
in time as they interact with a PTC, in order to preserve momentum.

Figure 5: (a) Example of a (finite) layered PTC. (b) Band structure of a PTC,
showing k-gaps and Zak numbers of the different bands. (c) Change of Zak phase
across the first k-gap. (d) Example of an edge between two PTCs with opposite
Zak phase. (e) Numerical demonstration of the effect of a temporal edge
between two topologically inequivalent PTCs: the parametric amplification
process is interrupted and the wave amplitude depleted for a few periods
following the temporal edge, forming a temporally localized state. Figures
reproduced from Ref. 69.

Let us consider an infinite series of switching events between
$\varepsilon_{1}$ and $\varepsilon_{2}$, as shown for a few cycles in Fig.
5(a). Being the problem periodic, the Floquet (Bloch) theorem holds, so that
the solution may be written as $\psi(t)=e^{i\Omega t}\phi(t)$, where
$\phi(t)=\phi(t+T)$, and $T=2\pi/\Omega$ is the period of the modulation. For
the specific case of a step-like modulation, the temporal scattering
coefficients can be related to each other by exploiting the symmetry
$T_{12}+R_{12}=T_{21}+R_{21}=1$, so that the Floquet bands can be calculated
analytically [69], and can be seen in Fig. 5(b). Notice how the k-gaps hosting
the amplifying states open at $\omega=\Omega/2$ (i.e. $\omega T=\pi$, where
$T$ is the modulation period). The reason for this can be read off the
solution for the double-switching problem in Eq. 15. Notice how the duration
$\Delta t_{2}$ determines whether or not gain will occur: if the duration of
the modulation is half of the period $\Delta t_{2}=T/2$, and we assume
$\omega_{1}\approx\omega_{2}=\omega$, then at each switching process a net
energy input from the contribution of the $\sin^{2}(\omega T/2)$ term will be
coherently fed into the waves, leading to exponential growth or, in in the
present of dominant losses, loss-compensation [70]. Related ideas were
recently investigated, such as “multilayer” temporal slabs to design temporal
transfer functions, in analogy with multilayer matching filters in spatial
transmission line theory [71, 72], and the effects of modulating both
dielectric and magnetic parameters in a PTC [73].

###  3.1 Topology in photonic time-crystals

One recent direction emerged in the context of spatial crystals is that of
topologically nontrivial photonic phases and symmetry-protected edge modes
found at their interface with a topologically inequivalent crystal. The
natural question of whether such a framework exists for temporal crystals has
recently been answered in the affirmative, although experimental observations
of temporal topological edge states are still missing [69]. In 1D (infinite)
crystals the topological character of the system is quantified via the Zak
phase. A temporal edge state must therefore lie at the temporal interface
between PTCs characterized by different Zak phases. For a layered PTC, the Zak
phase can be calculated analytically via Floquet theory, and its change across
the first k-gap is shown in Fig. 5(c) [69]. Far from a trivial generalisation,
however, a temporal edge is a markedly different object than a spatial one.
The edge modes commonly found at the interface between topologically
inequivalent crystals are now to be sought near (more specifically later than)
a specific instant of time, at which the properties of the PTC are suddenly
changed. In addition, while spatial edge states occur within the frequency gap
of a material where the eigenstates of the infinite crystal are evanescent,
temporal edge states in a PTC are only found within k-gaps, so that the
underlying wave dynamics is parametric amplification, and the transient wave
which constitutes the edge state has the effect of temporarily opposing the
exponential growth, as shown in Fig. 5(d-e) [69].

In spite of time being a one-dimensional quantity, time-modulation can also
provide a pathway towards higher-dimensional topological effects. This can be
accomplished by constructing synthetic frequency dimensions, by using the
frequency spectrum of the modes supported by a structure as an effective
lattice of states, in analogy with the sites of a tight-binding lattice, as
illustrated in Fig. 6(a-c).

Figure 6: (a) The resonant modes of a structure form a lattice. (b) Hopping
through its sites can be enabled by time modulation, which can couple their
different frequencies. (c) The resulting Hamiltonian is analogous to that of
electrons in a periodic ionic potential. (d) A ring resonator combined with a
modulator can reproduce such a model, (e) enabling hopping between its equally
spaced resonant modes [74]. (f-h) A synthetic dimension may be constructed in
a tight-binding model by introducing long-term coupling between the elements
of a chain. In the time-domain, this can be realized by modulating the system
at multiple frequencies [75]. (i-h) A spatially 1D chain of ring resonators
can be combined with temporal modulation to form a synthetic 2D lattice [76].
Figures adapted from Refs. 76, 75, 74.

Similarly to electrons in a simple one-dimensional lattice, photons in e.g. a
periodically modulated ring resonator may be described by a tight-binding
Hamiltonian:

$\displaystyle
H_{1}=g\sum_{m}(a_{m+1}^{\dagger}a_{m}e^{i\phi}+a_{m}^{\dagger}a_{m+1}e^{-i\phi})$
(16)

where $a_{m}^{\dagger}$ and $a_{m}$ are the photon creation and annihilation
operators for the $m^{th}$ mode on the lattice, $g=\kappa\Omega/2\pi$ is the
hopping amplitude, $\kappa$ is the modulation strength, $\Omega$ is the
modulation frequency and $\phi$ is the modulation phase [74]. In fact, just as
electrons hop between sites of a lattice, a dynamical modulation can make
photons hop between different modes of the resonator, as long as the
modulation frequency is similar to the difference $\Omega_{R}=2\pi v_{g}/L$
between the modes of the structure, where $v_{g}$ and $L$ are the group
velocity of the waveguide constituting the resonator and $L$ is its length.
The simplest instance of such a lattice may be realized with a single ring
resonator combined with a periodic modulator, shown schematically in Fig. 6.

Importantly, such a ladder does not need to be unidimensional: additional
dimensions may be constructed from any degree of freedom by simply introducing
additional coupling terms between a lattice of resonators, or, in a synthetic
frequency dimension, between modes [76], which means that a linear lattice can
now be equivalently described as a folded one to highlight its higher
dimensionality, as illustrated in Fig. 6(f-h). With time-modulation, this can
be done by considering two periodic modulations of different frequencies e.g.
$\Omega_{1}=\Omega_{R}$ and $\Omega_{N}=N\Omega_{R}$, thus introducing
coupling between $N^{th}$-neighbouring modes in the lattice so that the
Hamiltonian above can be generalized to:

$\displaystyle
H_{N}=g\sum_{m}(a_{m+1}^{\dagger}a_{m}e^{i\phi}+h.c.)+g_{N}\sum_{m}(a_{m+N}^{\dagger}a_{m}e^{i\phi_{N}}+h.c.)$
(17)

where $g_{N}=\kappa_{N}\Omega_{R}/2/pi$. Here, the second term accounts for
the coupling between modes separated by a frequency gap $N\Omega_{R}$, and it
is mathematically equivalent to the addition of a second dimension. Time-
modulation can also be combined with other degrees of freedom, such as orbital
angular momentum [77] or spatial lattices [76] to implement a range of
photonic effects, from non-trivial Chern numbers to effective gauge fields and
spin-momentum locking. Implementations of synthetic frequency dimensions have
been realized in RF with varactors [78] and superconducting resonators [79],
and in the near-infrared with fiber and on-chip electro-optic modulators [80,
81], as well as all-optically via four-wave mixing, with frequency spacings
generally ranging from MHz to THz [74]. A recent proposal makes use of a
single ring-resonator coupled to an array of optomechanical pumps and a single
electro-optic modulator to realize an on-chip 2D topological insulator with
two synthetic dimensions [82].

###  3.2 Non-Hermiticity and disorder

Another promising direction for new opportunities in wave manipulation is the
interplay between temporal structure and Hermiticity. Gain in electromagnetism
has conventionally been associated with the use of active media, providing
wave amplification in the form of a negative dissipation. However, as
discussed above, time-modulation offers an alternative route towards gain.
Nevertheless, a complete analogy between non-Hermitian gain/loss and
parametric processes cannot be drawn: in fact, it is evident from our previous
discussion that a mere time-modulation cannot provide loss, as a consequence
of momentum conservation, whereas an imaginary component in the response
parameters of a medium can be used to provide either gain or dissipation
depending on its sign.

Figure 7: (a) Two resonators coupled via a combination of constant ($k_{0}$),
time-dependent Hermitian ($\Delta k^{\prime}$) and time-dependent non-
Hermitian ($\Delta k^{\prime\prime}$) coupling terms. (b-e) Wave amplitude
under excitation via Port 1 in the two resonators (orange and blue) for two
arbitrary values of $\Delta k^{\prime}$ at the excitation frequency
($\omega_{1}$) and the two sidebands for (b) $k_{0}\neq 0$ and $\Delta
k^{\prime\prime}=0$ (energy stored in both sidebands and both resonators); (c)
$\Delta k^{\prime\prime}=0$ and $k_{0}=0$ (energy channeled to sideband
frequencies is all in the second resonator while the first only hosts the
input frequency); (d) $\Delta k^{\prime}=\Delta k^{\prime\prime}$ and
$k_{0}\neq 0$ (energy gained is nonreciprocally channeled into the upper
sideband and it is distributed over both resonators); (e) $\Delta
k^{\prime}=\Delta k^{\prime\prime}$ and $k_{0}=0$ (energy gained is
nonreciprocally channeled only into the upper sideband, and uniquely in the
second resonator). [83] (f) A random sequence of temporal $\delta$-kicks
results in (g) universal statistics observed by the energy in the system $U$.
The energy at the N-th kick $U_{N}$ is plotted against the step number $N$.
[84]. Figures adapted from Refs. 83, 84.

This distinction between parametric and non-Hermitian gain makes it natural to
ask what more can be achieved by modulating in time the non-Hermitian
component of the material response. As discussed in Sec. 2, the switching of
dissipation not only impacts the amplitude of the wave, but also its phase,
and can also generate backward waves. In fact, as recently shown, non-
Hermitian time-modulation can be used for nonreciprocal mode-steering in
frequency space [83]. Consider the setup shown in Fig. 7a: two resonators with
mode-lifetimes $\gamma_{1}$, $\gamma_{2}$ are coupled via a time-dependent
coupling constant $k_{0}+\Delta k^{\prime}\cos(\Omega t)+j\Delta
k^{\prime\prime}\sin(\Omega t)$, resulting in a non-Hermitian time-Floquet
Hamiltonian. Normally a periodic modulation of the system enables similar
upconversion and downconversion between modes of a structure. However, it was
shown that the inclusion of a non-Hermitian component in the time-modulation
of the coupling constant between the resonators can hijack such mode coupling,
rerouting all of the energy into one of the two resonators, thus producing
nonreciprocal gain [83].

The link between time-modulation and parity-time (PT) symmetric Hamiltonians
has also been discussed in some depth for the parametric resonance case, where
the frequency of the modulation doubles that of the incoming light [85].
Elegantly, PT symmetry breaking in these parametric systems can be related to
the stability conditions of the Mathieu equation governing a harmonically
modulated structure [86]. Such time-modulation-induced PT symmetric physics
additionally enables not only lasing instabilities, but also bidirectional
invisibility due to anisotropic transmission resonance, when two PT-symmetric
time-Floquet slabs are combined together such that their respective parametric
oscillations cancel out exactly. New perspectives relating exceptional point
physics and parametric modulation may be also found in Refs. 87, 88, 89,
whereas Ref. 90 investigates topological phases in non-Hermitian Floquet
photonic systems.

Another interesting direction for non-Hermitian time-modulated systems relates
to the temporal analogue of the causality relations of conventional passive
media [45]. As a well-known consequence of causality, imposed by demanding
that the response function of a system is only non-zero at times following an
input signal, the real and imaginary parts of the response of a system must
relate to each other through the Kramers-Kronig relations. However, the
spectral analogue of these relations can also be constructed: if one would
like the spectral response of a system to be uniquely non-zero for frequencies
higher or lower than the input frequency, such that only upconversion or
downconversion occurs, then there exist temporal Kramers-Kronig relations
which can provide a recipe for how the Hermitian and the non-Hermitian
components of a time-modulation must be related, in order for such asymmetric
frequency conversion to occur. Related concepts have also been exploited for
event-cloaking, broadband absorption and synthetic frequency dimensions [45].

The introduction of temporal disorder constitutes yet another avenue where the
inequivalence between space and time may bear new perspectives in long-
standing problems such as Anderson localization [91, 84, 92, 93].
Interestingly, however, the conventional localization observed in spatially
disordered media does not occur in a time-modulated system. Instead, in a
similar fashion as the occurrence of edge states, the onset of the temporal
analogue of Anderson localization manifests itself as an exponential growth in
the energy of the waves. Such wave dynamics has been investigated both
analytically [84], demonstrating the universal statistics occurring in
$\delta$-kicked temporal media, and numerically [92], as a disordered
perturbation in a PTC. Experiments on wave dynamics in temporally disordered
systems have also recently been performed with water waves [91].

In the following section, we move on to briefly discuss temporally structured
surfaces, whereby the interplay of spatial and temporal discontinuities can
enable not only several realistic platforms for implementations with
metasurfaces, but also completely new regimes of wave scattering.

###  3.3 Time-modulated surfaces

Figure 8: (a) Light scattered from a flat, time-varying metasurface can access
a locus of states which form a hyperbola in phase space, as opposed to an
ellipse in spatial metasurfaces. (b) Implementation of a time-modulated
metasurface at GHz frequencies. (c) A 1 MHz modulation performs efficient
frequency conversion through time-modulation [94]. (d) Illustration of
surface-wave excitation via Wood anomalies using a spatial (e) and temporal
(f) modulation of a surface (g). (h) Example of Wood anomaly observed in
transmission, the surface-wave excitation is explicitly shown in (i) from a
Finite-Element-Time-Domain simulation [95]. Figures adapted from [94, 95].

Amidst the rise of 2D materials, spatially structured surfaces, known as
metasurfaces, have recently attracted enormous interest as the ultrathin
counterpart of metamaterials. Periodically structured surfaces, or gratings,
have however occupied a central role in photonics since its early days. In its
interaction with waves in the far-field, a 1D-grating can couple waves whose
dispersion lives on the edge of the light cone by trading in-plane ($k_{x}$)
and out-of-plane ($k_{z}$) momentum via both its in-plane structure and its
transverse boundary, such that they must add in quadrature to the total
momentum $k_{0}=\varepsilon\omega/c_{0}$ of the impinging waves.
Interestingly, time-modulated surfaces enable sharply different wave dynamics
due the fact that the light cone treats time on a different footing that any
of the spatial dimensions: if we consider light impinging on a flat, time-
modulated surface, its in-plane momentum $k_{x}$ will now be conserved,
whereas $k_{z}$ and, importantly the wave frequency $\omega$, are now allowed
to change. Taking a vertical cut of the light cone for a fixed value of
$k_{x}$, we see that the two quantities being traded by the surface, $k_{z}$
and $\omega$ must not add, but rather subtract in quadrature. As a result, the
locus of points described by the available modes in the far-field is no longer
elliptical, but hyperbolic. One consequence is that an increase in frequency
caused by a temporal modulation corresponds to an increase in out-of-plane
momentum and vice versa, the opposite of what happens between momentum
components as they interact with a spatial grating.

One direction for time-modulated surfaces is the opportunity to couple
radiation to surface waves in the absence of any surface structure. Since
their dispersion curve lies outside of the light cone, surface waves are
typically excited by breaking translational symmetry along the surface either
via a localized near-field probe like a SNOM tip [96], or periodically using a
grating which induces a Wood Anomaly [97]. However, the argument above can be
exploited to envision surface-wave excitation via a temporal grating [95], as
depicted in Fig. 8(d-i). This approach brings the advantage of complete
reconfigurability and use of higher-quality pristine materials, in particular
in highly tunable 2D materials such as graphene and other Van der Waals
polaritonic materials such as hexagonal boron nitride and MoO3, thereby
circumventing the very need for surface fabrication or near-field excitation
techniques.

Another key technological application of time-modulated surfaces is that of
frequency conversion [98]. The advent of metasurfaces across the
electromagnetic spectrum has led to a surge of opportunities for advancing
this known field of research both theoretically [99, 100, 101, 102, 103, 104,
105, 106, 107] and experimentally [94, 108, 109, 110, 111]. Recent
implementations of frequency translation with metasurfaces have been recently
carried out in the microwave regime [94], and implementations with
experimentally more practical discretized arrays of time-and spacetime-
modulated reactive elements are in steady development [102, 109]. Several more
experimental implementations are discussed in Sec. 5. Interestingly, the
combination of bianisotropy and time-modulation has also been predicted to
give rise to nonreciprocal response (see Sec. 4.1) 112. Specific scattering
studies have also been dedicated to periodically modulated slabs [113],
particularly in the context of parametric amplification [107].

This recent surge of interest in time-modulated media has certainly
highlighted the fundamental nature of the novel wave phenomena that time-
modulation can unlock, and it is likely that a wealth of opportunities still
lies unexplored for even the simplest spatially homogeneous systems. More
opportunities involving surface structures have been explored in the context
of space-time metasurfaces (Sec. 4.4). However, before discussing in detail
the several new related concepts and implementations, it is instructive to
extend the theoretical background presented so far to the case of space-time
modulations, which introduce a wealth of additional phenomenology to the
scattering processes discussed so far.

##  4 Spatiotemporal metamaterials

Spatiotemporal modulations generally imprint a travelling-wave-like
perturbation, of the form $\delta\varepsilon(x,t)=\delta\varepsilon(gx-\Omega
t)$ (and similarly for $\mu$) onto the response parameters of a medium, where
$g=2\pi/L$ and $\Omega=2\pi/T$ are the spatial and temporal components of the
reciprocal lattice vector associated with the spatial and temporal periods $L$
and $T$ of the modulation, which form a spatiotemporal lattice vector
$p=(T,L)$, as depicted in Fig. 9(a-b) [114]. The idea of travelling-wave
modulation has been in use since the early days of travelling-wave amplifiers,
whereby electron beams are used to pump energy into co-propagating microwave
beams [115], and have undergone a number of investigations in the 60’s [116,
117, 118, 119, 120], and later in the early 2000’s [121, 122]. Recently, the
interest in these systems has revamped, with intense research efforts in the
modelling and realization of space-time modulated nonreciprocal systems (see
Sec. 4.1) [123, 124, 125, 126, 127, 128, 25]. In space-time media, neither
frequency nor momentum are individually conserved, but rather a spatiotemporal
Bloch vector $(\omega,k)$ forms a good quantum number, and the fields can
generally be expressed as a superposition of a discrete set of Floquet-Bloch
modes characterized by frequency-momentum vectors $(k+ng,\omega+n\Omega)$,
where $n\in\mathbb{Z}$ as shown in Fig.9(c-f), so that an eigenmode will have
the form:

$\displaystyle\psi(x,t)=e^{j(\omega t-kx)}\sum_{n}a_{n}e^{jn(\Omega t-gx)}$
(18)

and an eigenvalue problem may be set in terms of either $\omega(k)$ or
$k(\omega)$. Importantly, these new reciprocal lattice vectors $(g,\Omega)$
are not generally horizontal (c), but may form an arbitrary angle in
$\omega-k$ space, which may be smaller than the slope $c_{0}$ of the bands for
the background medium (subluminal, panel d) or larger (superluminal, panel e),
reducing to a pure temporal modulation in the limit $v_{m}=\Omega/g\to\infty$
as the entire medium is then effectively modulated instantaneously (panel f).
Interestingly, a peculiar regime of modulation velocities:

$\displaystyle\frac{c_{0}}{\sqrt{(1+\max(\delta\varepsilon))}\sqrt{(1+\max(\delta\mu))}}<v_{m}<\frac{c_{0}}{\sqrt{(1-\max(\delta\varepsilon))}\sqrt{(1-\max(\delta\mu))}}$
(19)

exists, where the Fourier expansion in Eq. 18 does not converge, as first
shown by Cassedy in Ref. 118. We discuss this exotic _luminal_ regime [129] in
detail in Sec. 4.3. In the next section we discuss the role of space-time
modulation for the engineering of nonreciprocal scatterers, which first
motivated the interest in this field.

Figure 9: (a) Spacetime diagram of a space-time crystal with alternating
refractive indices $n_{i}$ and $n_{j}$, spatiotemporal lattice vector $p$ and
velocity $v_{m}$. (b) Double-period space-time crystal, characterized by two
different lattice vectors $p_{A}$ and $p_{B}$, and two different velocities
$v_{m1}$ and $v_{m2}$ [130]. (c) The band structure for a spatial crystal with
$\varepsilon\sim\cos(gz-\Omega t)$ becomes asymmetric as (d) a finite temporal
component $\Omega$, and therefore a finite modulation velocity
$v_{m}=\Omega/g$, is introduced. This is caused by reciprocal lattice vectors
(dashed arrows) acquiring a non-zero frequency component. The angle formed by
the reciprocal lattice vectors with the slope of the bands determines a
subluminal (c-d) and a superluminal (e-f) regime [129]. (g) Nonreciprocal
mode-coupling in a space-time modulated waveguide: the modulation can couple
forward modes between each other as their frequencies and momenta are matched
by the space-time modulation, while two backward waves are not coupled, so
that an impinging backward wave is effectively unchanged [123]. Figures
adapted from 130, 129, 123.

###  4.1 Compact nonreciprocal devices without magnetic bias

The rise of space-time media was largely fuelled by the quest for magnet-free
nonreciprocity that has dominated the metamaterials scene of the past decade.
Nonreciprocity is the violation of the Lorentz reciprocity theorem, by which
swapping source and receiver leaves the scattering properties of a medium
unaltered [123]. Nonreciprocal components such as isolators and circulators
are crucial for duplex communications, enabling simultaneous transmission and
reception at the same frequency on the same channel without parasitic
reflections, as well as other applications such as laser protection from back-
scattering and field enhancement. The common approach to breaking this
symmetry of space is the use of gyromagnetic media, where the presence of a
magnetic field breaks time-reversal symmetry. One common nonreciprocal effect
encountered in basic electromagnetism is Faraday rotation, whereby the
rotation of the polarization of a wave passing through a gyromagnetic medium
depends on the direction from which the waves are impinging on the medium. One
key drawback of gyromagnetic media, however, is their requirement of strong
magnetic fields and large footprint for such effects to be appreciable,
leading to bulky components, incompatible with CMOS technology. Time-
modulation offers the opportunity of explicitly breaking time-reversal
symmetry, thereby violating reciprocity without the use of strong magnetic
fields [123].

Figure 10: (a) Angular momentum biasing: a directional modulation along a ring
resonator splits the degeneracy between clockwise and anti-clockwise
travelling states, (b) in analogy with the Zeeman splitting of electronic
states in a magnetic field [131]. (c) This can be realized by discretizing the
elements of the ring, e.g. periodically modulating three strongly coupled
resonators, with a phase of $120\deg$ between them [132]. (d) Example of
operation of a non-reciprocal ring resonator coupled to two waveguides:
excitation from the bottom (top) of the channel waveguide (on the right of the
ring) excites a counterclockwise (clockwise) mode, whose resonant frequency
has been offset from the one of the clockwise (counterclockwise) mode, so that
the direction of excitation determines the efficiency of the coupling to the
resonator (e), and hence of the transmission to the output port of the channel
waveguide [131]. (f-g) Theoretical and (h) experimental performance of an RF
circulator made with angular-momentum biased resonators: panels (f) and (g)
show the reciprocal response in the absence (f) and presence (g) of angular
biasing, while panel (h) shows the performance of the experimental
implementation of the circulator [132]. Figures reproduced from Refs. 131,
132.

The basic idea of using ST modulation for optical isolation may be summarized
in Fig. 9(g): the modulation produces transitions between two optical states
which differ in both momentum $k$ and frequency $\omega$: as a result, if the
system hosting the waves (e.g. a waveguide) supports multiple bands, to a pair
of forward-propagating modes coupled by the ST modulation may not correspond a
pair of backward-propagating ones. Therefore, while a forward incoming wave at
frequency $\omega_{1}$ may be completely converted into a new mode with
frequency $\omega_{2}$, a backward wave at $\omega_{1}$ will not be converted
as it will be mismatched in frequency and/or momentum. Hence, by introducing a
narrowband filter at $\omega_{2}$, the system allows propagation only in the
backward direction, thus achieving optical isolation [123, 126], in a similar
fashion to the way a diode conduces electricity unidirectionally [133]. An
equivalent way of viewing this scattering asymmetry in an extended medium is
the opening of an asymmetric band gap in the photonic dispersion of a
material, allowing propagation along one direction only [134].

Ring-resonators constitute perhaps the most promising components for
electromagnetic nonreciprocity. The idea of using ring resonators for
nonreciprocity makes use of the concept of angular momentum biasing [Fig.
10(a)]: a circular, directional bias splits the degeneracy between clockwise
and anticlockwise states, effectively mimicking the Zeeman splitting of atomic
physics [Fig. 10(b)]. In practice, this can be achieved with e.g. three
elements periodically modulated with a $120\deg$ phase difference between them
[Fig. 10(c)]. Let us consider the two-port system formed by two linear
waveguides coupled to the biased ring resonator shown in panel d. Thanks to
the splitting between clockwise and anticlockwise states, if the channel
waveguide (on the right of the ring in the panel) is excited at a frequency
resonant with the anticlockwise mode from the bottom port, its power will be
mainly rerouted by the ring resonator into the drop channel (left of the
ring). On the contrary, a wave impinging from the top of the waveguide, which
would normally couple to the clockwise state would now be off-resonance, thus
coupling poorly to the mode of the resonator, and being largely transmitted to
the bottom port of the channel waveguide [131]. This strategy can be used to
realize electromagnetic circulators, as shown in Fig. 10(f-h). A circulator
consists of three ports, and its purpose is to enable transmission from port 1
to port 2, port 2 to port 3 and port 3 to port 1, while impeding transmission
in the opposite sense (i.e. $2\to 1$, $3\to 2$ and $1\to 3$). Figure 10(f)
shows the transmission from channel 1 to 2 and one to 3 in the absence of
angular-momentum bias, which is perfectly symmetric. Once the bias is turned
on, backward propagation is forbidden, as shown in simulations [Fig. 10(g)]
and experimental data [Fig. 10(h)] [132]. Other electromagnetic
implementations have been realized with silicon waveguides [127], as well as
microstrip transmission lines [124], with modulation frequencies ranging from
hundreds of MHz to tens of GHz.

Multiple extensive reviews on space-time modulation focusing on nonreciprocity
have recently been published, so we refer the reader to them for further
details, and move on to illustrate further opportunities for space-time media
[123, 135].

###  4.2 Synthetic motion with spacetime media

The interaction of waves with moving bodies is at the origin of a myriad of
physical phenomena, such as the Doppler effect, the Fresnel drag by which a
moving medium drags light, or even the generation of hydroelectricity [18,
136, 137, 138]. Moreover, the electromagnetic response of a moving system is
inherently nonreciprocal, as it is associated with a broken time-reversal
symmetry [139, 140, 141]. In fact, flipping the arrow of time also requires
flipping the velocity of all the moving components, leading thereby to a
distinct optical platform. The nonreciprocity and the non-Hermitian nature of
the electromagnetic response of moving matter can enable unidirectional light
flows [141], classical and quantum non-contact friction [142, 143], parametric
amplification [144, 145, 141, 146], amongst others [3].

Unfortunately, it is impractical to take technological advantage of many of
the features highlighted in the previous paragraph, because a realistic value
for the velocity of a moving body is many orders of magnitude smaller than the
speed of light. Interestingly, the actual physical motion of a macroscopic
body may be imitated by a drift current in a solid state material with high-
mobility [147, 148, 149, 150]. For example, it was recently experimentally
verified that drifting electrons in graphene can lead to nonreciprocity at
terahertz frequencies and to a Fresnel drag [151, 152]. A different way to
realize an effective moving response without any actual physical motion is
through a travelling wave spacetime modulation [117, 118, 119, 114, 153, 154].
Such a solution is discussed next in detail.

Figure 11: (a) and (b) A synthetic spacetime modulated crystal (panel a) is
formally equivalent to a fictitious moving bianisotropic time-invariant
crystal (panel b) when $v_{m}<c_{0}$. The velocity of the equivalent moving
crystal is identical to the modulation speed $\bf{v_{m}}$. (c) In the long
wavelength limit the system response can be homogenized and the crystal
behaves as a uniaxial dielectric moving with speed $\bf{v_{D}}$. The sign of
$v_{D}$ is not necessarily coincident with the sign of $v$. (d) Dispersion
diagram of the effective medium $\omega\,\,{\rm{vs}}\,\,k_{x}$. When only one
of the material parameters is spacetime modulated the dispersion is symmetric,
and the equivalent drag velocity vanishes (blue lines). When both
$\varepsilon$ and $\mu$ are spacetime modulated, the dispersion is asymmetric
(red lines, assuming $v_{D}>0$). The waves that propagate in the direction of
$\bf{v}_{D}$ have a larger group velocity than the waves that propagate in the
opposite direction [153]. (e) Drag velocity $v_{D}$ and (f) effective
bianisotropic coefficient $\xi_{\rm{ef}}$ as a function of the modulation
speed $v_{m}$ for a representative spacetime modulated crystal: the velocity
of the equivalent moving medium and the magneto-electric coupling coefficient
flip sign in the transition from the subluminal to the superluminal regime.
Adapted from 155.

A system featuring a travelling-wave-type modulation is characterized, in the
dispersionless limit, by the following constitutive relations:

$\left({\begin{array}[]{*{20}{c}}{{\bf{D}}}\\\
{{\bf{B}}}\end{array}}\right)=\left({\begin{array}[]{*{20}{c}}{\varepsilon\left({x-{v_{m}}t,y,z}\right){\bf{1}}}&{\bf{0}}\\\
{\bf{0}}&{\mu\left({x-{v_{m}}t,y,z}\right){\bf{1}}}\end{array}}\right)\cdot\left({\begin{array}[]{*{20}{c}}{{\bf{E}}}\\\
{{\bf{H}}}\end{array}}\right)$ (20)

where $v$ is the modulation speed [Fig. 11(a)]. This form of modulation in
space and time imparts a synthetic motion to the material response along the
$x$-direction with uniform speed $v_{m}$. This property implies that with a
suitable coordinate transformation one can switch to a frame where the
material response is time-invariant. In fact, a Galilean transformation of
coordinates,

$x^{\prime}=x-{v_{m}}t,\quad y^{\prime}=y,\quad z^{\prime}=z,\quad
t^{\prime}=t,$ (21)

preserves the usual structure of the Maxwell’s equations, with the transformed
fields related to the original fields as follows ${\bf{D^{\prime}}}={\bf{D}}$,
${\bf{B^{\prime}}}={\bf{B}}$,
${\bf{E^{\prime}}}={\bf{E}}+{\bf{v_{m}}}\times{\bf{B}}$,
${\bf{H^{\prime}}}={\bf{H}}-{\bf{v_{m}}}\times{\bf{D}}$ and
${\bf{v_{m}}}={v_{m}}{\bf{\hat{x}}}$. In the new coordinates, the constitutive
relations are time-independent

$\left({\begin{array}[]{*{20}{c}}{{\bf{D^{\prime}}}}\\\
{{\bf{B^{\prime}}}}\end{array}}\right)=\left({\begin{array}[]{*{20}{c}}{{\bf{\hat{\varepsilon}^{\prime}}}\left({{\bf{r^{\prime}}}}\right)}&{{\bf{\hat{\xi}^{\prime}}}\left({{\bf{r^{\prime}}}}\right)}\\\
{{\bf{\hat{\zeta}^{\prime}}}\left({{\bf{r^{\prime}}}}\right)}&{{\bf{\hat{\mu}^{\prime}}}\left({{\bf{r^{\prime}}}}\right)}\end{array}}\right)\cdot\left({\begin{array}[]{*{20}{c}}{{\bf{E^{\prime}}}}\\\
{{\bf{H^{\prime}}}}\end{array}}\right),$ (22)

with ${\bf{r^{\prime}}}={\bf{r}}-{\bf{v_{m}}}t$ and the transformed effective
parameters given by

$\displaystyle{\bf{\hat{\varepsilon}^{\prime}}}$
$\displaystyle=\varepsilon\left({\frac{1}{{1-\varepsilon\mu{v_{m}^{2}}}}{{\bf{1}}_{t}}+{\bf{\hat{x}}}\otimes{\bf{\hat{x}}}}\right),$
(23) $\displaystyle{\bf{\hat{\mu}^{\prime}}}$
$\displaystyle=\mu\left({\frac{1}{{1-\varepsilon\mu{v_{m}^{2}}}}{{\bf{1}}_{t}}+{\bf{\hat{x}}}\otimes{\bf{\hat{x}}}}\right),$
(24) $\displaystyle{\bf{\hat{\xi}^{\prime}}}$
$\displaystyle=-{\bf{\hat{\zeta}^{\prime}}}=\frac{{-\varepsilon\mu}}{{1-\varepsilon\mu{v_{m}^{2}}}}{\bf{v_{m}}}\times{\bf{1}}.$
(25)

In the above,
${{\bf{1}}_{t}}={\bf{\hat{y}}}\otimes{\bf{\hat{y}}}+{\bf{\hat{z}}}\otimes{\bf{\hat{z}}}$
and $\otimes$ represents the tensor product of two vectors. As seen, the
coordinate transformation originates a bianisotropic-type coupling determined
by the magneto-electric tensors $\bf{\hat{\xi}^{\prime}}$ and
$\bf{\hat{\zeta}^{\prime}}$ [156], such that the electric displacement vector
$\bf{D^{\prime}}$ and the magnetic induction $\bf{B^{\prime}}$ depend on both
the electric and magnetic fields $\bf{E^{\prime}}$ and $\bf{H^{\prime}}$.
Indeed, as further discussed below, one of the peculiar features of the
travelling-wave spacetime modulation is that it mixes the electric and
magnetic responses, leading to the possibility of a giant bianisotropy in the
quasi-static limit [155, 153].

The new coordinate system is not associated with an inertial frame. An
immediate consequence of this property is that the vacuum response does not
stay invariant under the coordinate transformation (21). Usually, this does
not create any difficulties, but it is relevant to mention that it is also
possible to obtain a time-invariant response with a Lorentz coordinate
transformation. Such a solution is restricted to subluminal modulation
velocities, and thereby the Galilean transformation is typically the preferred
one.

Interestingly, the constitutive relations (22) in the new coordinate system
are reminiscent of those of a moving dielectric medium [139]. Due to this
feature the electrodynamics of spacetime media with travelling wave
modulations bears many similarities to the electrodynamics of moving bodies
[153]. Yet, it is important to underline that a spacetime modulated dielectric
crystal is not equivalent to a _moving_ dielectric crystal. In other words,
impressing a time modulation on the parameters of a dielectric photonic
crystal is not equivalent to setting the same photonic crystal into motion. A
moving dielectric crystal would have a bianisotropic response in a frame where
its speed $v$ is nontrival, very different from the constitutive relations
(20). Below, we revisit this discussion in the context of effective medium
theory.

Even though the spacetime modulated dielectric system is not equivalent to a
moving dielectric, its response can be precisely linked to that of a
fictitious moving medium in the subluminal case. In fact, as previously noted,
a suitable Lorentz transformation makes the response (20) of a spacetime
system independent of time, analogous to Eq. (22). Thus, the original
constitutive relations (20) are indistinguishable from those of a hypothetical
moving system with a bianisotropic response in the co-moving frame of the type
(22). In other words, the synthetic motion provided by the travelling wave
modulation imitates the actual physical motion of a fictitious time-invariant
bianisotropic crystal [Fig. 11(b)].

The previous discussion is completely general, apart from the assumption of a
travelling wave modulation. In the following, we focus our attention on
periodic systems and in the long wavelength regime. Effective medium methods
have long been used to provide a simplified description of the wave
propagation in complex media and metamaterials [157, 158, 159, 160]. The
effective medium formalism is useful not only because it enables analytical
modeling of the relevant phenomena, but also because it clearly pinpoints the
key features of the system that originate the peculiar physics.

The standard homogenization approach is largely rooted on the idea of “spatial
averaging”, so that the effective response describes the dynamics of the
envelopes of the electromagnetic fields [18]. In particular, in the
traditional framework, there is no time averaging. Due to this reason,
standard homogenization approaches are not directly applicable to spacetime
crystals, where the microscopic response of the system depends simultaneously
on space and on time.

At present, there is no general effective medium theory for spacetime
crystals. Fortunately, the particular class of spacetime crystals with
travelling wave modulations can be homogenized using standard ideas [155, 161,
162]. In fact, since a travelling wave modulated spacetime crystal is
virtually equivalent to a moving system, it is possible to find the effective
response with well established methods by working in the co-moving frame
(primed coordinates) where the medium response is time invariant [155].

To illustrate these ideas, we consider a one-dimensional photonic crystal such
that the permittivity $\varepsilon$ and permeability $\mu$ are independent of
the $y$ and $z$ coordinates. It is well known, that for stratified systems the
components of the $\bf{E}$ and $\bf{H}$ fields parallel to the interfaces ($y$
and $z$ components) and the components of the $\bf{D}$ and $\bf{B}$ fields
normal to the interfaces ($x$ components) are constant in the long wavelength
limit [163], i.e., when the primed field envelopes vary sufficiently slowly in
space and in time. Taking this result into account, one can relate the
spatially averaged $\left\langle{{\bf{D^{\prime}}}}\right\rangle$ and
$\left\langle{{\bf{B^{\prime}}}}\right\rangle$ with the spatially averaged
$\left\langle{{\bf{E^{\prime}}}}\right\rangle$ and
$\left\langle{{\bf{H^{\prime}}}}\right\rangle$ [155]. The effective parameters
in the original (laboratory) frame can then be found with an inverse Galilean
(or Lorentz) transformation. Such procedure leads to the following
constitutive relations in the lab frame:

$\left({\begin{array}[]{*{20}{c}}{\left\langle{\bf{D}}\right\rangle}\\\
{\left\langle{\bf{B}}\right\rangle}\end{array}}\right)=\left({\begin{array}[]{*{20}{c}}{{{{\bf{\hat{\varepsilon}}}}_{{\rm{ef}}}}}&{{\xi_{{\rm{ef}}}}{\bf{\hat{x}}}\times{\bf{1}}}\\\
{-{\xi_{{\rm{ef}}}}{\bf{\hat{x}}}\times{\bf{1}}}&{{{{\bf{\hat{\mu}}}}_{{\rm{ef}}}}}\end{array}}\right)\cdot\left({\begin{array}[]{*{20}{c}}{\left\langle{\bf{E}}\right\rangle}\\\
{\left\langle{\bf{H}}\right\rangle}\end{array}}\right),$ (26)

where ${\bf{\hat{\varepsilon}}}_{\rm{ef}}$, ${\bf{\hat{\mu}}}_{\rm{ef}}$, and
$\xi_{\rm{ef}}$ are some parameters that can be written explicitly in terms of
the permittivity and permeability profiles $\varepsilon(x)$ and $\mu(x)$
[155]. It turns out that when only one of the material parameters is modulated
in spacetime the magnetoelectric coupling coefficient $\xi_{\rm{ef}}$ vanishes
and the effective medium behaves as a standard uniaxial dielectric [155, 153].
For example, if the material permeability is independent of space and time,
then $\xi_{\rm{ef}}=0$ and ${\bf{\hat{\mu}}}_{\rm{ef}}=\mu{\bf{1}}$, for any
permittivity profile $\varepsilon(x)$. This property can be intuitively
understood by noting that when $\mu=const.$, the “microscopic” constitutive
relation ${\bf{B}}=\mu{\bf{H}}$ implies that the averaged fields are also
linked by
$\left\langle{\bf{B}}\right\rangle=\mu\left\langle{\bf{H}}\right\rangle$,
which corresponds to a trivial effective magnetic response. Note that this
argument is only justified in the static limit, as in the dynamical case the
second order spatial dispersion effects may lead to artificial magnetism [164,
158].

Remarkably, when both $\varepsilon$ and $\mu$ are modulated in spacetime,
$\xi_{\rm{ef}}$ can be nontrivial. In other words, notwithstanding that at the
microscopic level there is no magnetoelectric coupling [Eq. (20)], the
effective medium is characterized by a bianisotropic response in the static
limit. While it is not unusual that the complex wave interactions in a
metamaterial without inversion symmetry can result in a magnetoelectric
coupling [156], having a nontrivial bianisotropic response in the long
wavelength limit is a rather unique result 111For completeness, we point out
that some antiferromagnets, such as chromium oxide, may be characterized by an
intrinsic axion-type bianisotropic response in the static limit [165, 166]..
In fact, in typical time-invariant systems the electric and magnetic fields
are decoupled in the static limit. Thereby, the electric and magnetic
responses of any conventional metamaterial are necessarily decoupled in the
static limit and there is no bianisotropy. In contrast, for a spacetime
modulated system the electric and the magnetic fields are never fully
decoupled, as the time modulation of the material implies always some
nontrivial time dynamics. The strength and sign of $\xi_{\rm{ef}}$ can be
tuned by changing the modulation speed and the profiles of the permittivity
and permeability. As illustrated in Fig. 11(f), $\xi_{\rm{ef}}$ can be
exceptionally large for modulation speeds approaching the speed of light in
the relevant materials. This results in giant nonreciprocity, as has also been
shown through Floquet-Bloch expansions [167]. The transition from the
subluminal to the superluminal regime is marked by a change in sign of
$\xi_{\rm{ef}}$. Furthermore, it can be shown that constitutive relations (26)
are identical to those of a fictitious moving anisotropic dielectric
characterized in the respective co-moving frame by the permittivity
${{{\bf{\hat{\varepsilon}}}}_{{\rm{D}}}}$ and by the permeability
${{{\bf{\hat{\mu}}}}_{{\rm{D}}}}$. The equivalent medium moves with a speed
$v_{D}$ with respect to the lab frame. The parameters
${{{\bf{\hat{\varepsilon}}}}_{{\rm{D}}}}$, ${{{\bf{\hat{\mu}}}}_{{\rm{D}}}}$,
and $v_{D}$ are determined univocally by ${\bf{\hat{\varepsilon}}}_{\rm{ef}}$,
${\bf{\hat{\mu}}}_{\rm{ef}}$, and $\xi_{\rm{ef}}$[155]. In general, $v_{D}$
may be rather different from the modulation speed $v$, both in amplitude and
sign. The velocity $v_{D}$ can be nonzero only if $\varepsilon$ and $\mu$ are
both modulated in spacetime, i.e., only if $\xi_{\rm{ef}}\neq 0$. Therefore,
it follows that a spacetime modulated photonic crystal can imitate precisely
the response of a moving dielectric in the long wavelength limit. Remarkably,
the velocity of the equivalent moving medium can be a very significant
fraction of $c_{0}$ [Fig. 11(e)]. Thus, spacetime modulated systems are ideal
platforms to mimic on a table-top experiment the electrodynamics of bodies
moving at relativistic velocities, the only caveat being that some degree of
frequency mixing will, in general, occur upon propagation in a space-time
medium.

In particular, similar to a moving medium, a spacetime modulated crystal can
produce a drag effect [153]. Specifically, a wave co-propagating with the
equivalent moving medium, i.e. towards the direction
${\rm{sgn}}\left({{v_{\rm{D}}}}\right){\bf{\hat{x}}}$, moves faster than a
wave propagating in the opposite direction, due to the synthetic Fresnel-drag
effect [138]. A clear fingerprint of the Fresnel drag can be detected in the
dispersion of $\omega$ vs. $k$ of the electromagnetic modes of the spacetime
crystal. In fact, when both $\varepsilon$ and $\mu$ are modulated in time, the
slopes of the dispersion of the photonic states near $k=0$ depends on the
direction of propagation of the wave. The dispersion of the waves that co-
propagate with the equivalent moving medium exhibits a larger slope than the
dispersion of the modes that propagate in the opposite direction, as depicted
in Fig. 11(d). Interestingly, these slopes are predicted _exactly_ by the
discussed effective medium theory, and thereby the homogenization theory is
expected to be rather accurate and useful to characterize excitations that
vary sufficiently slowly in space and in time. Furthermore, the effective
medium theory is exact for all frequencies when the parameters of the crystal
are matched: $\varepsilon/\mu=const.$. In fact, the dispersion of a matched
photonic crystal is generally linear for all frequencies. However, as we show
in the next section, even this statement can be violated in an exotic class of
spacetime media recently termed _luminal_ metamaterials, which we discuss in
the next section.

###  4.3 Luminal amplification and spatiotemporal localization

Figure 12: (a) Electric field intensity at the output port vs time transmitted
through different thicknesses (blue to red) $d$ of luminal crystal. (b) In the
gain region the permittivity gradient is positive, and vice versa. The gain is
maximum at the points (in time or space) where the velocity of the waves
matches the velocity of the grating. (c-d) Power spectra of the transmitted
waves for input frequency (c) $\omega_{0}=8\Omega$, with $\Omega$ the
modulation frequency, and (d) for a DC input field ($\omega_{0}=0$), showing
the frequency generation responsible for the overall gain [129, 168]. (e-f)
Trajectories of a point of constant phase along the spacetime variable
$X=x-v_{m}t$. (e) outside of the localization regime (resulting in
oscillations which periodically compress and decompress the pulse, panel g),
and (f) within the localization regime, resulting in the amplification and
compression described in (a-d) [169]. Figures adapted from Refs. 168, 129,
169.

If the speed of a space-time modulation falls within the modulation velocity
regime in Eq. 19 (the extent of this _luminal_ regime increases with the
modulation amplitude), the relevant wave physics changes dramatically,
entering a very peculiar unstable phase whose underlying mechanism is,
however, completely distinct from the parametric amplification processes
described in Secs. 2 and 3 [129]. In this regime, all forward bands become
almost degenerate, as the frequency-momentum reciprocal lattice vectors
effectively align with the forward bands. This regime marks a transitions
between the subluminal and superluminal regimes in Fig. 9(d) and (e)
respectively. Due to the resulting strong degeneracy between forward-
travelling waves, the eigenmodes of these luminal systems cannot be written in
Bloch form [169]: the effect of the modulation, in fact, is to couple all of
the forward bands, such that an impinging wave would emerge as a
supercontinuum, or frequency comb. Such a transmission process is shown in
Fig. 12(a-d): the waves are compressed into a train of pulses, and both the
total energy of the system and its compression increase exponentially with
respect to the propagation length (or time) in the luminal medium, as well as
the amplitude and rate of the modulation. In real space, this amplification
process can be viewed as the capturing of field lines by the synthetically
moving grating [170]. The wave dynamics in a luminal medium is captured by the
approximate equation for the energy density $U$ (which becomes exact if
$\varepsilon=\mu$ everywhere):

$\displaystyle\frac{\partial\ln{U}}{\partial
t^{\prime}}=\frac{(v_{m}+c_{l})}{2}\bigg{[}\frac{\partial\ln\mu}{\partial
X}+\frac{\partial\ln\varepsilon}{\partial
X}\bigg{]}+(v_{m}-c_{l})\frac{\partial\ln{U}}{\partial X}$ (27)

where $v_{m}$ is the grating velocity, $c_{l}(X)=(\varepsilon\mu)^{-1/2}$ and
$X=x-v_{m}t$ and $t^{\prime}=t$ are moving coordinates, which can be solved by
successive iterations [168]. The characteristic feature of the luminal regime
is the presence of points along the grating where the local phase velocity
$c_{l}(X)$ of the waves matches the velocity $v_{m}$ of the grating (crossings
between the black, dashed horizontal line and the green sinusoidal grating
profile in Fig. 12(b)). The presence of these velocity-matching points implies
that the lines of force are trapped within each period of the grating, merely
subject to the first term, which acts as an energy source term, resulting from
the spatiotemporal change in $\varepsilon$ and $\mu$. In fact, these local
gradients in the electromagnetic parameters (first term in Eq. 27) will pump
energy into the waves, or deplete them, based on their sign, whereas the phase
velocity of the waves in the region adjacent to the velocity-matching points
determines whether these will function as attractors ($\pi/2$ in Fig. 12(b))
or repellors ($3\pi/2$ in Fig. 12(b)) as a result of the Poynting flux driving
power towards or away from them[171]. It has been noted that this completely
linear mechanism results in pulse compression or expansion [172]. On the other
hand, as the modulation velocity approaches the edges of the luminal regime, a
localization transition takes place, before which the response of the system
is characterized by strong oscillations in time consisting of periodic field
compression and expansion [169], as shown in figure 12 (e-g). Importantly,
luminal amplification does not rely on back-scattering, as opposed to
parametric amplification, and in fact it occurs just as well in a system which
is impedance-matched everywhere in spacetime, in sharp contrast to parametric
amplification, which relies on time-reversal as explained in Sec. 3.

###  4.4 Space-time metasurfaces

Most implementations of time-varying and space-time-varying systems are hard
to achieve in bulk, as any modulation mechanism, e.g. a pump pulse, would
generally need to impinge on the system from an additional direction. In
addition, field penetration into a bulky structure is generally inhomogeneous,
so that in practice only a small surface layer of a bulk medium would
effectively be modulated. These considerations, combined with with the ease of
fabrication of metasurfaces compared to bulk metamaterials, have led to a
concentration of research interest in space-time metasurfaces, whereby a
travelling-wave modulation is applied to the surface impedance of a thin
sheet, with proposals covering a wide range of ideas and applications, some of
which are shown in Fig. 10.

Figure 13: Examples of applications of space-time metasurfaces: (a) An
isolator based on the unidirectional excitation of evanescent modes; (b) Waves
scattering off a STM experience a nonreciprocal phase shift; (c) A STM-based
circulator [99]. (d) Illustration of an experimentally realized space-time
coding metasurface, consisting of a square array of individual voltage-
controlled elements with binary impedance values, enabling encoding and
imprinting of arbitrary phase and amplitude modulation onto different
scattered harmonics for (e) beam-steering and shaping [109]. (f) Illustration
of wave power combining using STMs: incoming waves with frequency $\omega_{0}$
coming from a discrete range of different angles can be scattered towards the
same outgoing direction by imprinting the necessary frequency-wavevector
shifts via the STM. (g) A proposal of an angular momentum-biased metasurface
for generation of orbital angular momentum states: each azimuthal section is
temporally modulated with a different relative phase, imparting the necessary
angular-momentum bias. Figures adapted from Refs. 99, 109, 173

Systems such as isolators [Fig. 13(a)], nonreciprocal phase shifters [Fig.
13(b)] and circulators [Fig. 13(c)] using space-time modulated metasurfaces
were designed, extending magnet-free nonreciprocity to surface implementations
[106, 174, 175, 99]. A promising idea to exploit these concepts in practice,
at least at low frequencies, has come in the form of space-time-coding digital
metasurfaces [Fig. 13(d)]: these systems consist of arrays of elements whose
impedances are designed to take a discrete set of values, so that the overall
response of the structure can be encoded as an array of bits (if only two
values exist for each element) [176]. Switching the individual impedances of a
voltage-controlled array of such elements can enable exquisite control over
the metasurfaces response in a system whose design can be systematically
described using information theory concepts to tailor shape and direction of
electromagnetic beams [Fig. 13(e)] [109], as well as to perform photonic
analog computing [177]. Furthermore, close to the luminal regime,
opportunities for nonreciprocal hyperbolic propagation [154], as well as
vacuum Čerenkov radiation [178] using space-time metasurfaces have recently
been highlighted.

Another application of space-time metasurfaces that was recently proposed is
that of power combining of waves, a long-standing challenge in laser
technology. Here the strategy consists of using a spatiotemporal modulation of
a surface to effectively trade the difference in angle of incidence between a
discrete set of incoming waves with identical frequency in exchange for a
difference between frequencies of the outgoing waves, which emerge, at the
same angle, and at an equally spaced set of frequencies, as illustrated in
Fig. 13(f) [103]. More exotic ideas include angular-momentum-biased
metasurfaces [Fig. 13(g)], which can impart a geometric phase on the impinging
waves, and engineer photonic states with finite orbital angular momentum and
optical vortices [173, 105]. Finally, the concept of space-time metasurfaces
has recently been imported into the quantum realm, promising full control of
spatial, temporal and spin degrees of freedom for non-classical light, with
several opportunities for the generation of entangled photonic states, thereby
anticipating a whole new scenery for this field to expand towards [179].

###  4.5 Further directions

Among new directions for space-time media, dispersive effects offer additional
knobs for mode-engineering [180], for which much still remains to be explored.
Interestingly, spatial dispersion may be effectively engineered into a
material via time-modulation, just as temporal dispersion arises from spatial
structure [181]. Chiral versions of synthetically moving and amplifying
electromagnetic media, mimicking the Archimedean screw for fluids, have
recently been proposed, and may be realized with circularly polarized pump-
probe experiments [182]. Topology also occupies a prime seat, with Floquet
topological insulators having been realized in acoustics [183], and
theoretically proposed for light [184]. Further related opportunities have
been demonstrated for photonic realizations of the Aharonov-Bohm effect [185],
as well as higher dimensional topological effects such as Weyl points
(topological invariants for 3D structures) in spatially 2D systems, realized
by exploiting a synthetic frequency dimension [186]. The idea of space-time
media has also been formalized in the interesting mathematical formulation of
“field patterns”. These objects arise from the consecutive spatial and
temporal scattering in a material that features a checkerboard-like structure
of its response parameters in space and time, and may offer new angles to
study space-time metamaterials in the future [187, 188, 162]. Finally, beside
photonic systems, non-electromagnetic dynamical metamaterials have already
acquired significant momentum, with several theoretical proposals [189, 190,
191] and experimental realizations, including asymmetric charge diffusion
[192], nonreciprocity [140, 193, 194], Floquet topological insulators [183]
and topological pumping [195] amongst many others. In particular, digitally
controlled acoustic meta-atoms have recently attracted significant interest to
probe time-modulation physics in acoustics [196, 197].

Having discussed the wealth of potential new physics enabled by time-varying
media, we now turn our attention to the latest technological advances towards
experimental implementations in photonics, and their related challenges.

##  5 Experimental advances and challenges in all-optical implementations

The experimental realisation of time-varying effects constitutes a new, broad
and fruitful field of research across the wider wave-physics and engineering
spectrum. Implementation in optics, and more particularly in nanoscale
architectures and metasurfaces offers new opportunities to address both
fundamental open questions in wave physics as well as material science, and
potentially groundbreaking technological pathways, such as low-footprint
ultrafast optical modulators and nonreciprocal components. This is
particularly challenging due to the need for modulating a medium on a temporal
scale similar to the optical frequency of the light field. Time-varying
effects have been demonstrated in mechanical [198], magnetic [199], acousto-
optic [200], opto-mechanical [201] and electronic systems [127]. In
particular, electromagnetic metasurfaces can be modulated via mechanical
actuation, chemical reactions, or phase-change materials, as well as
electrically [111]. Nevertheless, their slow modulation speed compared to
near-optical timescales makes these poor candidates for achieving sizable
frequency shifts and other time-varying effects in the visible or near-
infrared ranges. The following section focuses on experimental realisations of
time-varying media in these optical frequency ranges. Emphasis is placed on
metasurfaces as these have subwavelength thicknesses that allow for purely
temporal modulations of the medium. In a metasurface, the various constraints
of bulk media such as loss or self-broadening are avoided, and the propagation
of the modulating or probe beam in the medium does not need to be taken into
account. We will review in a first step the state of the art of time-varying
photonics experiments using nonlinear optical processes in semiconductors,
more specifically their most common implementations: epsilon near zero (ENZ)
materials, hybrid ENZ platforms and high-index dielectrics. We will then
discuss emerging paths to implement all-optical ultrafast modulation within
new structures or materials such as quantum wells, magnetic materials,
multilayered metamaterials and 2D materials.

###  5.1 Photocarrier excitation and nonlinear optical modulation

Thanks to their wide variety of implementations and frameworks, as well as
their ultrafast nature, nonlinear optical interactions have proven to be a
fertile ground for the development of time-varying systems at optical
frequencies. In particular, modulation effects originated by short laser
pulses interacting with a nonlinear medium occur on sub-ps timescales, well
beyond the reach of electro-optic modulation, and very close to the time of an
optical cycle (1-10 fs regime). Optical modulation of a medium stems from a
change in $\varepsilon$, the relative permittivity, due to the nonlinear
response in a material. In semiconductors, these effects are often driven by
out-of-equilibrium electronic populations, called hot-electrons. Nonlinear
effects in semiconductors arise because of electron excitation of either real
or imaginary states. Transitions through real states are realized by
photocarrier excitation, that is intraband or interband transitions of the
electrons within the active medium. The redistribution of electrons within the
valence and/or conduction bands will affect the permittivity and thus the
refractive index of the material on timescales limited either by the rate of
transfer of energy from the light beam to the electrons or by the light pulse
duration. This allows for sub-ps modulation of $\varepsilon$ in a variety of
media of interest for photonics, including dielectrics and metals. Transitions
through virtual states are on the other hand much faster, but lack the
strength of photocarrier excitations: nonlinear processes via virtual states,
such as four-wave-mixing and sum-frequency-generation, usually exhibit low
efficiencies. Both photocarrier excitation and nonlinear optical modulation
induce ultrafast modulation of $\varepsilon$, generally limited by the pulse
duration and not material-limited. In the next section we will discuss recent
progress in nonlinear optical modulation for time-varying effects in two main
platforms: epsilon-near-zero media and high-index dielectrics.

####  5.1.1 Epsilon-Near-Zero materials, transparent conducting oxides and
nonlinear optics

ENZ materials exhibit various interesting properties around their ENZ
frequency, where the real part of their permittivity crosses zero, while the
imaginary part stays relatively low. ENZ media feature guided and plasmonic
modes for field enhancement, slow-light effects in waveguides and phase-
matching relaxation [202]. Of particular interest are transparent conducting
oxides (TCOs), degenerately doped semiconductors whose band structures allow
for ENZ frequencies in the near-IR with strong optical nonlinearities. Indium
Tin Oxide (ITO) and Aluminium Zinc Oxide (AZO) are well known, silicon-
technology compatible TCOs with very strong nonlinear optical response [203,
204, 205, 206] which makes them good candidates as platforms for time-varying
metasurfaces. Particularly, intraband transitions in ITO lead to a strong
redistribution of the effective mass of conducting electrons and thus an
efficient shifting of the plasma frequency in the material’s Drude dispersion
as shown in Fig. 14(a). As shown by Alam et al. in 2016 [203], order of unity
modulation of the refractive index can be achieved in ITO. Keeping in mind
that for an instantaneous change of index, the frequency shift due to time-
refraction is proportional to $\Delta n/n$, where $n$ is the refractive index
and $\Delta n$ the index change, strong frequency shifts can be achieved in
ITO due to its low refractive index near its ENZ frequency, even for small
changes $\Delta n$.

Time-refraction in pump probe experiments has been shown in bulk ITO and AZO
thin-films [207, 208, 209, 210, 211], with frequency shifts as strong as 58
THz [209] when the probe field and the modulation overlap [see diagram in Fig.
14(b)]. Time-refraction not only applies to the reflected and transmitted
probe pulse, but also to its phase conjugation and negative refraction, as
shown in Fig. 14(c-f). In the works of Ferrera et al. [207], Vezzoli et al.
[208] and Bruno et al. [209], negatively refracted and phase-conjugated
signals were recorded simultaneously with the time-refracted signal, with the
internal efficiencies going above unity for these ultrafast, purely time-
varying signals [208]. This demonstrated the potential of these TCOs for time-
varying applications, yet the strength of the modulation and signal was mostly
achieved thanks to a significant propagation within the medium, and was
hampered by losses in the medium (for ITO, the penetration depth of light
around the ENZ frequency is about 100 nm). Time-refraction in a 80 nm thick
ITO metasurface was demonstrated in Liu et al.’s work [212] in 2021, which in
turn highlighted the need for a new understanding of the role of saturation of
photocarrier excitation in the modulation of permittivity as large field
intensities are now confined to significantly smaller portions of space. In
this work it was shown that, as expected, a shorter pulse will lead to a
stronger shift in frequency due to a larger $\mathrm{d}{n}/\mathrm{d}{t}$
during the short propagation time within the ITO slab, but more interestingly
that saturation happens at lower energies for longer pulses. This indicates
that the ultimate material response time for short and intense modulation
constitutes still an open research question.

Nanostructuring ENZ materials is a powerful strategy to increase the optical
field enhancement and lead to efficient ultrafast modulation as demonstrated
by Guo et al.’s work [213] with ITO nanorods. In this work, the thickness of
the medium is 2.6 $\mu$m, a length sufficient to cause significant loss upon
propagation. An alternative to nanostructuring the ITO to increase the field
enhancement consists of exploiting the plasmonic mode exhibited by flat ENZ
thin films [214]. In Bohn et al.’s work [215], the plasmonic ENZ mode was
excited using a Kretschmann configuration, leading to a change in reflectance
of 45%.

While ENZ materials allow for large nonlinear responses and time-modulation,
their large refractive index mismatch with air or larger-than-one-index media
calls for advanced photonic architectures to ensure efficient coupling as we
will address in the next section.

Figure 14: (a,b) Concept of photocarrier excitation and time-refraction in ITO
[210]: (a) A pump pulse transfers energy to the electrons in the conduction
band, leading to a temporal variation in the refractive index. (b) As the
probe experiences different indices at different delays, its spectral content
will be shifted (redshift for negative delay, blueshift for positive). (c-f)
Measured time-refraction, phase conjugation and negative refraction signals
from a 500 nm thick AZO slab pumped by 105 fs pulses, at an energy of 770
GW/cm2in a degenerate pump-probe experiment at 1400 nm [209]. (g) Strong
coupling between a plasmonic antenna and an ENZ thin film [216]: electric
field $|E|^{2}$ distribution obtained from FDTD for an ITO layer thickness of
40 nm and Au antenna length of 400 nm. (h) Resulting field distribution as a
function of metasurface depth. (i-k) Probe spectrum at various delays for a
central wavelength of 1304 nm for 50 fs pulses in a strongly-coupled plasmonic
antenna-ENZ system [217]. The increase in modulating pump intensities (0.5, 1
and 2 GW/cm2) leads to a shift near zero delay. Figures adapted from Refs.
210, 209, 216, 217.

####  5.1.2 Hybrid platforms for time-varying experiments

To compensate for the lack of interaction volume in a metasurface, good in-
coupling and enhancement of the electric field is necessary to achieve strong
nonlinear optical modulation. ENZ material-only metasurfaces lack strong
resonant behaviour, and this has prompted the community to include additional
elements to achieve efficient time modulation. Well-understood and easy-to-
realize, plasmonic-ENZ metasurfaces provide light coupling and local field
enhancement to otherwise impedance-mismatched ENZ materials.

Plasmonic nanoantennas couple efficiently propagating radiation to their near-
field. When plasmonic resonances are excited, electromagnetic hot-spots form
in the surrounding media. For this reason, metallic and more particularly Au
nanoantennas have been used to couple light from the far-field to a thin ENZ
substrate, where the strong field enhancement leads to efficient nonlinear
modulation of the medium. In addition, strong coupling of localized plasmon
resonances in Au antennas with the plasmonic modes of an ENZ film can be
achieved [216] (see Fig. 14(g-h)). The coupled antenna-ENZ thin film enhances
the field in the ENZ medium, and strong frequency shifts originating from
time-refraction, as well as efficient negative refraction, have been measured
from ITO films [216, 218, 212] with a record 11.2 THz frequency shift being
recorded at a comparatively low power of 4 GW/cm2in Pang et al.’s work [217]
[Fig. 14(i-k)]. Such time-varying metasurfaces are at the moment limited by
the low damage threshold of the plasmonic antennas, the damage threshold of
ENZ materials such as ITO or AZO being much higher.

In order to circumvent damage threshold constraints, architectures where the
optical field is localized in the ENZ layer only have been explored: Au films
can be used as a perfectly conducting layer in the near-IR and IR, beneath the
ENZ thin film to increase its coupling to free space. Even though this is an
impedance-matching effect, it can be understood as a superposition of the
suppression of reflection for p-polarized light at the Brewster angle and the
suppression of transmission from the reflective layer. Such a system was used
by Yang et al. [219]: a layer of In:CdO with an ENZ frequency at 2.1 $\mu$m
exhibited large nonlinearities when illuminated at its Brewster angle, with
the 9th harmonic being generated and measured from the metasurface. Time-
varying effects are noticeable in the harmonic spectrum, with for example the
5th harmonic exhibiting two peaks: a small one at $5f$ and a large, time-
refracted one 48 THz below arising from the modulation of the index by the
pump and the shifted frequencies being upconverted (that is a shift of about 2
THz of the probe signal due to time-refraction).

Hybrid plasmonic-ENZ systems thus present an improvement in realising time-
varying metasurfaces using the favorable properties of ENZ materials and other
TCOs. This calls for further investigation of the implementation of ENZ
properties in other types of metasurfaces, such as the multilayered systems
presented in section 5.2.3.

Figure 15: (a) Schematic of an Si metasurface engineered to support collective
high-Q Fano resonances [220]. (b) Schematic of the self-induced blue shift of
harmonics in the resulting Si nanoantenna cavity: photons in the antenna will
undergo a blueshift due to the rapidly changing permittivity of the medium
before being upconverted via third-harmonic generation. (c) Measured spectrum
evolution for a 80 fs pump pulse length in a GaAs high-$Q$ metasurface [221].
The fringes are measured on only one side of the spectrum due to the breaking
of time-reversal asymmetry caused by the modulation. (d) Schematic of a
nonreciprocal Si metasurface [222]: the travelling wave modulation breaks the
space-time symmetry of the reflection phase change. Figures adapted from Refs.
220, 221, 222

####  5.1.3 High-index dielectric metasurfaces as time-varying media

In the quest for resonant-enhanced materials for time-modulation, high-index
dielectrics provide an alternative to ENZ structures, as they offer
possibilities for various resonant architectures and a high damage threshold.
More in particular, nanoantennas with Mie resonances or bound states in the
continuum allow for strong time-modulation by combining efficient coupling to
the active medium and a good field enhancement. That is because even though
$\Delta n/n$ is low in a high-index dielectric, the change of phase of the
metasurface can be strong when the material is modulated near a resonance.
Shifts in the resonance of a photonic crystal [223] were demonstrated, as well
as in Mie-resonant systems, in a range of materials ranging from Si [224, 225,
226] to GaAs [227], GaP [228] and Ge [229]. GaAs and Si exhibited stronger
modulation efficiencies as they feature a direct band gap in the visible to
near-IR region, while Si relies on intraband transitions, less efficient for
this class of materials [227].

Time-refraction was observed both in Si [220] and GaAs [221] high-$Q$
nanoantenna metasurfaces. Shcherbakov et al. [220] reported a 8.3 THz shift in
the third harmonic signal from a Si metasurface with a collective Fano
resonance as shown in Fig. 15(a), i.e. around 2 THz shift in the self-
modulated pump signal. As depicted in Fig. 15(b) this is explained by the
upconversion to third harmonic of the time-refracted pump field, with the
change of index of the nanoantenna cavity originating from the pump itself as
well. This is a smaller shift than the best results achieved with combined
plasmonic antenna-ENZ systems, but comparable to what was achieved in bulk ITO
[212] and In:CdO on Au [219]. It is also worth noting that the system operates
at 11 GW/cm2, underlining the higher damage threshold of dielectric antennas
in comparison to plasmonic antennas.

Though high-index dielectric antennas exhibit higher damage threshold and
provide more flexibility thanks to the tunability of the resonances by
nanostructuring, as opposed to bulk ENZ properties, the interplay between the
width of the resonance and that of the modulating pulse must be considered
carefully. The higher the $Q$ factor, the stronger the potential for a
modulation as the field enhancement is higher at resonance, yet if the
duration of the modulating optical pulse is made shorter than the resonance
lifetime, in order to accelerate the modulation and obtain stronger time-
varying effects, its spectral content will exceed the resonance bandwidth and
only a modest portion will interact. This calls for nondegenerate pump-probe
experiments to show the full-potential of time-varying media: Karl et al.
[221] recorded the clear appearance of fringes in the probe spectrum due to
time-varying effects as shown in Fig. 15(c), by independently controlling the
pump pulse length via pulse chirping and the spectral content of the probe via
spectral filtering of a supercontinuum pulse.

One can also take advantage of the phase-shift induced by a high-index
nanoantenna array: Guo et al. [222] demonstrated nonreciprocal light
reflection in a Si nanobar metasurface. To this end, a spacetime modulation
was induced by the interference and beating of two pump beams with a 6 nm
difference in central wavelength and a nonreciprocal frequency shift was
measured from the reflection of the forward and backward-propagating probe
[Fig. 15(d)].

Thanks to their strong, tunable resonances, high-index dielectric metasurfaces
have proven to be a reliable platform for time-varying experiments. Although
third-order nonlinearities in these materials are weaker than in ITO or AZO,
and nanostructuring puts a lower cap on damage threshold and interaction
volume, these metasurfaces bring concurrence to ENZ media thanks to the
maturity of nanoantenna fabrication technology and the control of the
scattering phase enabled by such systems.

###  5.2 New leads for time-varying metasurfaces

It is clear that better and more efficient material platforms for time-varying
media are sought after and will likely appear in the near future, to allow for
more efficient modulations, larger bandwidth, and higher damage thresholds. In
this final section we identify a few promising candidates, namely quantum well
polaritons, magneto-optical modulation, multilayered ENZ metamaterials and 2D
materials, exhibiting the potential to enhance time-modulation effects and we
discuss their respective strengths and weaknesses.

####  5.2.1 Quantum well polaritons

Mann et al. [230] showed in 2021 the pulse-limited modulation of intersubband
polaritonic metasurfaces, using the coupling between patch antennas and
intersubband transitions in multi-quantum wells (MQWs). The antenna resonance
matches the transition dipole moment of the MQWs, which leads to Rabi
splitting at low intensities. As the intensity increases, the ground state is
depleted, which induces a change in coupling and absorption properties. The
system operates at intensities between 70 kW/cm2 and 700 kW/cm2, with a change
in absorption of order of unity at the antenna resonant frequency. Though the
change in reflection is quite weak at such powers (at best 8% here), it is
worth noting that the intensities here are much lower than those used in other
nonlinear optical experiments for time-modulation. On the other hand, the
recovery time is dictated by the relaxation time of the excited state of the
MQWs, here of 1.7 ps. In comparison, the modulation recovery time in
unsaturated ITO is 360 fs [231].

####  5.2.2 Magneto-optical modulation

Though the mutual effects of electric and magnetic fields are quite weak, one
can consider using the magneto-optical Kerr effect (MOKE) or Faraday effect
(MOFE) to achieve ultrafast time-dynamics and thus time-varying physics in a
magnetic medium. MOKE (MOFE) consists in the rotation of reflected
(transmitted) light by a magnetic field [232, 233]. A first stone was laid
when Beaurepaire et al. [234] showed in 1996 sub-ps switching of spin in
ferromagnets using a MOKE configuration. Ultrafast spin switching was
demonstrated without magnetic fields using a circularly-polarized optical
pulse [235], exploiting the dynamics of the inverse Faraday effect.
Particularly, Stanciu et al. [236] demonstrated a 40 fs all-optical switching
of magnetization. Though in this experiment the origin of the switching was
finally found out to be originating from the heating of the medium by the
laser pulses, Mangin et al. [237] later showed helicity-dependent switching in
various magnetic media, independently of the threshold switching temperature
of the medium. This opened a new path towards ultrafast magnetic memory
writing and other applications of optical switching of magnets.

Experiments involving the use of a magnetic field use a common figure of
merit, the $\delta$ parameter, defined as the percentage change of reflection
when the magnetisation is reversed. That is for a magnetic field M and a
reflection $R$, $\delta=[R(\textbf{M})-R(-\textbf{M})]/R(\textbf{0})$. This
value is usually quite low in bulk ferromagnetic thin films, ranging from
$10^{-5}$ to $10^{-3}$. Nanostructuring can help increase this figure of
merit: the coupling of surface plasmon polaritons from an Au grating to a
ferromagnetic substrate allows for a stronger MOKE effect [238], leading to a
$\delta$ parameter of up to 24 % [239]. In other works, plasmonic antennas in
a magnetic field were used to control the transmission of chiral light [240],
and dielectric nanoantennas were engineered via their electric and magnetic
Mie resonances to achieve large Faraday rotation [241]. Yet, these time-
modulations of the medium linked to the value of $\delta$ do require a
switching of the magnetisation, which happens on a much slower scale (100 ps
to ns [240, 242]) than the optical modulation from the MOKE itself. Hence, the
ultrafast aspect of the modulation may only come from the rotation of the
polarisation of light at a given, fixed magnetisation.

In a parallel stream of research, ultrafast spin currents have been excited in
magnetized thin films and heterostructures using optical pulses, paving the
way for THz emitter technologies [243, 244]. Photocurrents were generated on a
timescale of 330 fs in Huisman et al.’s work [245]. As the magnetic field does
not require switching and can be maintained constant, the modulation can be
considered as entirely optical and ultrafast. Qiu et al. demonstrated similar
spin current generation in an antiferromagnetic slab in the absence of
magnetic field [246], thanks to second-order optical nonlinearities. For a
more detailed review of THz emitters and wave generation the authors recommend
the review by Feng et al. [247]. While the potential of magneto-optical
effects for ultrafast phenomena has been demonstrated, their translation into
the framework of time-varying media remains unexplored, which calls for a new
push in investigations and experiments in this direction.

####  5.2.3 Multilayered metamaterials

Multilayered metal-dielectric structures are of great interest thanks to their
tunability to different regimes via fabrication, such as ENZ or hyperbolic. In
addition, their simple architecture for fabrication is well-understood from
effective-medium theory [248]. Artificial ENZ metamaterials exhibiting Dirac
cone-like dispersion [249] or multilayered thin films make valuable candidates
for time-varying experiments, as they exhibit the favorable ENZ physics, are
tunable and have the potential to exploit other meta-properties, e.g. good
coupling to far-field radiation. Sub-ps modulation of the effective
permittivity of a multilayered Au/TiO2 metamaterial was demonstrated by Rashed
et al. [250]. In parallel, enhancement of nonlinear properties around the ENZ
frequency of a multilayered Ag/SiO2 [251], as well as ultrafast modulation of
absorption [252] were reported. This enhancement is explained by the
dependence of the effective third-order nonlinear susceptibility on the
inverse of the medium’s effective index, which reaches a minimum in the
vicinity of the multilayered structure’s ENZ frequency. In a similar spirit,
ultrafast modulation of a metal-insulator-metal nanocavity was observed around
the low-energy ENZ point [253], but it was found that the switching speed was
limited by the carrier density and heat capacity of the metal. The authors
suggested the use of TCO-dielectric structures to accelerate the electron
dynamics of the process and achieve faster modulation. Alternatively,
multilayered ENZ metamaterials can be used to pin down and control the
plasmonic resonance of coupled antennas depending on the ENZ frequency [254],
with optical modulation allowing for control of the antenna resonance via the
multilayered substrate.

In a different paradigm, multilayered metal-dielectric antennas have been
engineered to support ultra-small mode volumes [255], and in this way couple
light efficiently to a WS2 monolayer [256]. WS2 and 2D materials are a
promising class of media for time-varying experiments as we discuss in the
following section. Although monolayers feature properties favorable to
modulation, their intrinsic atomic-scale thickness limits their coupling to
light: hyperbolic metamaterials such as a multilayered metal-dielectric
antenna can overcome this barrier and boost optical modulation. Furthermore,
efficient second and third harmonic generation was measured from multilayered
Au/SiO2 nanoantennas [257]. The efficiency of the second harmonic was proved
to originate from the multiple metal-dielectric interfaces rather than
symmetry-breaking, allowing for a polarisation-independent implementation of
multilayered antennas.

To sum up, multilayered ENZ metamaterials constitute promising candidates for
time-varying experiments as they exhibit ultrafast modulation and strong
nonlinear properties, while offering options for nanostructuring and resonance
engineering.

####  5.2.4 2D materials

A modern topic in nanophotonics, 2D materials exhibit many interesting
properties including high refractive index and ease of nanostructuring [258],
unconventional band structure and carrier dynamics such as Dirac cones [259],
and excitonic physics as well as ultra-high carrier mobility [260], resulting
in strong optical nonlinearities useful for material modulation [261, 262].
Transition metal dichalcogenides (TMDs), black phosphorus (BP) and graphene
feature sub-ps carrier excitation times [263, 264, 265] in single layer
structures. In particular, TMDs offer excitonic resonances with short
recombination times, while monolayer BP exhibits a tunable direct band gap
from the visible to the mid-IR. Additionally, BP’s anisotropic response has
also been shown to undergo modulation under laser excitation [266]. TMDs
exhibit a variety of useful properties for optical modulation [267], with
monolayer and low-dimensional TMDs exhibiting stronger nonlinear properties
than their bulk counterparts. In Nie et al.’s work [268], a 10 fs pulse of
about 1 GW/cm2was shown to excite photocarriers in 20 fs in low-dimensional
MoS2, though the change in transmission was only of 0.8%. Large second and
third-order nonlinearities have also been measured in low dimensional MoS2
[269, 270], MoSe2 [271], WS2 [272, 273] and WSe2 [274], as well as strong
tunability of the refractive index in WSe2, MoSe2 and MoS2 [275].
Additionally, the strong nonlinear properties in MoSe2 can be exploited thanks
to the comparatively high damage threshold of the monolayer as demonstrated by