ArticlePDF Available

Photonics of time-varying media

February 2022
Advanced Photonics 4(01)

February 2022
4(01)

DOI:10.1117/1.AP.4.1.014002

Authors:

Romain Tirole

CUNY Graduate Center

Shixiong Yin

City College of New York

Huanan Li

Nankai University

Show all 10 authorsHide

Time-varying media have recently emerged as a new paradigm for wave manipulation, due to the synergy between the discovery of highly nonlinear materials, such as epsilon-near-zero materials, and the quest for wave applications, such as magnet-free nonreciprocity, multimode 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 many 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. © The Authors. Published by SPIE and CLP under a Creative Commons Attribution 4.0 International License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.

Content uploaded by Romain Tirole

Content may be subject to copyright.

Photonics of time-varying media

Emanuele Galiffi ,a,b,*Romain Tirole ,b,†Shixiong Yin ,a,†Huanan Li ,a,c Stefano Vezzoli,bPaloma A. Huidobro ,d

Mário G. Silveirinha ,dRiccardo Sapienza ,bAndrea Alù ,a,e and J. B. Pendryb

aCity University of New York, Photonics Initiative, Advanced Science Research Center, New York, United States

bImperial College London, Blackett Laboratory, Department of Physics, London, United Kingdom

cNankai University, School of Physics, Tianjin, China

dInstituto de Telecomunicações, Instituto Superior Técnico-University of Lisbon, Lisboa, Portugal

eCity University of New York, Physics Program, Graduate Center, New York, United States

Abstract. Time-varying media have recently emerged as a new paradigm for wave manipulation, due to the

synergy between the discovery of highly nonlinear materials, such as epsilon-near-zero materials, and the

quest for wave applications, such as magnet-free nonreciprocity, multimode 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 many 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 media; temporal modulation; metamaterials; switching; optics; photonics; light.

Received Nov. 15, 2021; revised manuscript received Dec. 7, 2021; accepted for publication Dec. 8, 2021; published online

Feb. 14, 2022.

reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.

[DOI: 10.1117/1.AP.4.1.014002]

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.1Meanwhile, at cosmological scales, the measured ac-

celeration of the expansion of the universe makes it hard to

imagine a complete cosmological theory where temporal dy-

namics of, e.g., wave phenomena do not play a central role.

Yet, most basic physics is carried out under the assumption that

physical systems are passive, merely responding “after”any in-

put stimuli.

Occasionally, the need to account for certain truly multiphy-

sics or nonlinear phenomena brings up opportunities to develop

our understanding of wave physics in temporally inhomo-

geneous systems; for instance, to realize bench-top analogs

of relativistic phenomena, early attempts were made to tap into

time-modulated classical wave systems.2,3However, 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 con-

densed 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;4their very existence has been questioned and revised,5

and several increasingly successful implementations have re-

cently 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

*Address all correspondence to Emanuele Galiffi, egaliffi@gc.cuny.edu

†These authors contributed equally to this work.

Review Article

Advanced Photonics 014002-1 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

metamaterials.8In this pursuit, our need for even a basic under-

standing of wave phenomena in the time domain has been re-

flected in the wealth of surprising physical effects recently

unveiled, largely theoretically and in part experimentally, which

can be enabled by externally applying explicit temporal modu-

lations 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 pho-

tonic metamaterials in the hope that it may serve the photonics

and metamaterials community as a catalyst for current and fu-

ture explorations of this fascinating field. While not as detailed

as a tutorial paper, the review aims at presenting the basic build-

ing 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 2starts from the

basics of electromagnetic time-switching, expanding toward

more complex scenarios, such as as non-Hermitian and aniso-

tropic switching, temporal slabs, and space–time interfaces.

Section 3is dedicated to photonic time-crystals (PTCs), namely

electromagnetic systems undergoing infinite (periodic or disor-

dered) modulations of their parameters in time-only, and we

discuss some of its most promising directions toward, e.g., the

engineering of topological phases, synthetic frequency dimen-

sions, 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 intro-

duce the characteristic features of space–time band diagrams,

homogenization procedures for traveling-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 metasurfa-

ces and some of their applications. Finally, Sec. 5offers an

overview of the most promising platforms for experimental im-

plementations of time-modulation in photonics.

2 Wave Engineering with Temporal

Interfaces

Light scattering by spatial interfaces [Fig. 1(a)] is the basis of

many wave phenomena, from the simplest refraction effects to

the complexity of wave engineering in complex media involving

multilayers,9anisotropy,10 nonlinearity,11 extreme constitutive

parameters,12–14 or metamaterials and metasurfaces,15 to name

a few. How do electromagnetic waves behave when encounter-

ing 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 first investigated

by Morgenthaler in 1958.16 In parallel, the study of light inter-

actions 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 of time-switching in pho-

tonic 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 (Eand

H), must be finite at all points in space and time. Spatial boun-

dary conditions are therefore derived by integrating Maxwell’s

equations over an infinitesimal area and volume across the spa-

tial interface, implying the continuity of tangential Eand Hat 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 mac-

roscopic Maxwell’s curl equations:

∂B

∂t¼−∇×E;(1)

∂D

∂t¼−Jþ∇×H:(2)

Here, we assume that the medium is unbounded and homo-

geneous before and after a switching instance at t¼t0.

Integrating from t−

0¼t0−ϵto tþ

0¼t0þϵwith a ϵ→0þ,

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:

Bðt¼tþ

0Þ¼Bðt¼t−

0Þ;(3a)

Dðt¼tþ

0Þ¼Dðt¼t−

0Þ:(3b)

They ensure that the magnetic flux density Band the elec-

tric displacement Dvary continuously in the time domain.

(a) (b)

n1n1

n2n2

Fig. 1 Scattering at (a) a spatial discontinuity, generating trans-

mitted and reflected waves propagating in different media with

refractive indices n1and n2, 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 ω

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 kis con-

served, and the two scattered waves are both embedded in

the new medium.

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-2 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

Alternatively, the continuity of Band Dcan also be inter-

preted by the conservation of magnetic flux Φand electric

charges Q, as shown by Morgenthaler16 andAuldetal.,

respectively.

One of the most intriguing phenomena arising at a temporal

discontinuity is the wave scattering dual to that occurring at spa-

tial interfaces, producing forward and backward waves in time,

rather than waves reflected and transmitted in space.16,19–23 The

scattering coefficients can be found by applying the above tem-

poral boundary conditions. Consider a monochromatic plane

wave traveling in an unbounded, homogeneous, isotropic, non-

dispersive but time-varying medium. Both permittivity εand

permeability μare assumed to abruptly switch from their initial

values ε1and μ1to ε2and μ2at t¼t0. We can denote the in-

cident wave with Dx¼D1ejω1t−jkz and By¼B1ejω1t−jkz , where

ω1is the angular frequency and kis the wavenumber. The field

amplitudes are related by B1¼Z1D1with Z1¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

μ1∕ε1

p. The

fields after the switching event read

Dxðt>t

0Þ¼½Tejω2ðt−t0ÞþRe−jω2ðt−t0ÞD1ejðω1t0−k2zÞ;(4a)

Byðt>t

0Þ¼Z2½Tejω2ðt−t0Þ−Re−jω2ðt−t0ÞD1ejðω1t0−k2zÞ:(4b)

Tand Rare the transmission and reflection coefficients de-

fined for the electric displacement field, and Z2¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

μ2∕ε2

pis

the new wave impedance after switching. The temporal boun-

dary conditions [Eq. (3)] need to be satisfied everywhere in

space, which implies that k2¼k, or equivalently

ω2ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ε2μ2

p¼ω1ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ε1μ1

p:(5)

This condition ensures momentum conservation across tem-

poral interfaces, which is expected due to the preserved spatial

homogeneity.16,17,24,25 By equating the incident and scattered

Dand Bfields at t¼t0, we solve for the temporal scattering

coefficients:

T¼Z2þZ1

2Z2

;(6a)

R¼Z2−Z1

2Z2

:(6b)

The coefficients for the electric field can be easily found

by substituting the constitutive relation between Dand 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 as-

sumptions of spatial homogeneity, isotropy, and instantaneous

material response and may change if one or more of these

assumptions are relaxed.26 Illustrations of spatial and temporal

scattering processes are shown in Figs. 1(a),1(c) and 1(b),

1(d), respectively. As we consider more realistic and compli-

cated electrodynamic models of materials, such as anisotropy,

dispersion, and broken translational symmetry in space, the

temporal scattering is expectedly modified.27–33 To date, re-

search on these topics is very active, as discussed in the fol-

lowing sections.

2.2 Temporal Slabs

Leveraging scattering phenomena at time interfaces, research

efforts have been dedicated to the engineering of wave inter-

ference in time, for example, in the context of discrete time

crystals34–36 and time metamaterials.37 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 with spatial

interfaces. A key contrast between spatial and temporal scat-

tering 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 con-

secutive switching events. In Sec. 3, we discuss (quasi)periodi-

cally switched media.

The temporal interference introduced by a single temporal

slab was first investigated by Mendonça et al.38 Where they as-

sume that the refractive index of a medium is switched from n0

to n1, and then switched back to n0after a time interval τ. The

total transmission and reflection amplitudes were found to be

Ttotal ¼hcosðω1τÞþ j

2αð1þα2Þsinðω1τÞie−jω0τ;(7a)

Rtotal ¼−j

2αð1−α2Þsinðω1τÞejω0τ;(7b)

where α¼n0∕n1and ω0,ω1are 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.38 A tem-

poral Fabry–Perot slab with dispersion was recently studied in

Ref. 33. 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 a large amplification of the input energy

at a suitably tailored temporal slab. It can be seen from Eq. (7)

that such an amplification process attains the maximum level

when the time interval ω1τis an odd multiple of π∕2.On

the contrary, time reflections can be minimized with sinðωτ1Þ¼

0. This finding can be connected to the dual phenomenon

of impedance-matching layers in spatial scattering. Indeed,

Pacheco-Peña and Engheta39 recently introduced antireflection

temporal coatings by proposing to rapidly change the permittiv-

ity of an unbounded medium twice, 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 min-

imize the backward reflection when the permittivity is to be

switched from ε1to ε2, they introduce a temporal coating with

permittivity εeq and duration Δtin between. Impedance match-

ing is enabled with

εeq ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ε1ε2

p;Δt¼nTeq

4;n¼1;3;5;…(8)

where Teq ¼1∕feq is the equivalent period of the wave right

after the permittivity is switched to εeq and the equivalent

frequency feq ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ε1∕εeq

pf1. A numerical simulation is shown

in Fig. 2(b), indicating nearly vanishing reflection from the

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-3 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

temporal interfaces. Although Eq. (8) looks similar to the con-

dition for impedance matching in the spatial case,41 it is more

than a dual to its spatial counterpart. First, the frequency is

shifted from f1to f2¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ε1∕ε2

pf1, 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 en-

ergy manipulation can be achieved in these types of temporal

slabs once loss and gain are considered. In analogy to conven-

tional parity-time (PT) symmetry with balanced gain and loss,42

Li et al.40 investigated a scenario where a pair of temporal slabs

obey temporal parity-time (TPT) symmetry. In their work,

the temporal parity operation flips the arrow of time, whereas

the “time”operation is defined to reverse the waves in space.

TPT-symmetric temporal slabs are therefore realized if an un-

bounded medium is switched from a positive to a negative con-

ductivity in time, σ2¼−σ3>0, for equal duration Δtbefore

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.

By introducing a general temporal scattering matrix formu-

lation, which may be powerfully applied to any multilayer

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 sup-

ports unimodular eigenvalues. Interestingly, the nonorthogonal

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, PtotðtÞ¼PþðtÞþP−ðtÞwith PðtÞfor the forward/

backward propagating wave, can be significantly different

from the incident one. By controlling the relative phase of two

incident counterpropagating waves, one can reach drastically

different total power flows for the same TPT-symmetric tem-

poral bilayer. As an example, for n1¼1,n2¼2−j0.2,and

Fig. 2 (a) Schematic of the antireflection temporal coatings proposed in Ref. 39. (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 TPT symmetric structures pro-

posed in Ref. 40. (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) Maximum total power and (f) minimum total power. Insets are the field distributions in simu-

lations before, during, and after the TPT slabs. Figures adapted from Refs. 39 and 40.

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-4 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

2ω1Δt¼3.5, the temporal evolution of the normalized stored

energy density (solid curves) and its two components

Ui¼2ε½jEþðtÞj2þjE−ðtÞj2[dashed curves, proportional to

the total power flow in the two waves EþðtÞand E−ðtÞ,and

equal to the total stored energy in Hermitian media] and Uc¼

2Re½nðn−nÞEþðtÞE−ðtÞ(circle symbols, stemming from

the nonorthogonality of the counterpropagating waves in

non-Hermitian media) are shown in Figs. 2(e) and 2(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 func-

tion of the relative phase of the excitations, as a dual phenome-

non 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 in-

triguing possibility is to consider material anisotropy. In 2018,

Akbarzadeh et al.28 raised the question of whether it is possible

to realize an analog of Newton’s prism, shown in Fig. 3(a),ina

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 shown in Fig. 3(c),

in which an unbounded isotropic medium with scalar refrac-

tive index n1is switched to a uniaxial medium with n2¼

diagðn∥;n

∥;n

⊥Þat t¼t0. The right part of Fig. 3(c) represents

Fig. 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 frequen-

cies. (c) Implementation of the inverse prism proposed in Ref. 28. (d) Temporal aiming proposed in

Ref. 43. (e) The conventional Brewster angle. (f) Temporal Brewster angle described in Ref. 44.

Figures adapted with permission from Refs. 28,43, and 44.

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-5 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

the isofrequency curves of the medium before and after switch-

ing. Because the medium remains homogeneous, the conserva-

tion of the wavenumber k(momentum) still holds, leading to

“birefringence”at temporal frequencies. A monochromatic

wave at frequency ω1propagating in the isotropic medium be-

fore switching will then be mapped to different frequencies,

depending on its polarization and momentum, following:

n1ω1¼nðkÞ¼8

n∥ω2s-polarization

n∥n⊥ω2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

∥cos2θþn2

⊥sin2θ

pp-polarization ;(9)

with θ¼arc tan ðkz∕kxÞas 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 ω2for s- and p-polarizations.

One interesting phenomenon for wave propagation in aniso-

tropic crystals is that the group velocity does not necessarily

align with the phase velocity.10 Based on this feature, the con-

cept of temporal aiming was proposed in Ref. 43. Figure 3(d)

shows 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 εrðtÞ

shown in the insets would route the signal to different receivers

[denoted Rx 1 to 3 in Fig. 3(d)]. For an incoming wave with

propagation angle θ1¼tan−1ðkz∕kxÞ, the direction of the mo-

mentum kafter switching to the anisotropic medium remains

the same θ2k¼θ1k¼θ1, while that of the Poynting vector Sis

redirected to

θ2S¼arc tan εr2z

εr2x

tan θ1;(10)

assuming that the anisotropic medium is a uniaxial crystal with

ε2r¼diagðε2x;ε2x;ε2zÞ. The discrepancy between θ2Sand θ1

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 tem-

poral switching paves the way to routing waves through tempo-

ral interfaces, creating a form of temporal waveguiding.

Another interesting phenomenon exploiting temporal switch-

ing with anisotropic media is the temporal Brewster angle.44 The

conventional Brewster angle is defined as the angle at which the

reflection of the p-polarized incidence vanishes, as shown in

Fig. 3(e). Its temporal counterpart is shown in Fig. 3(f) and sim-

ilarly 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:

θtB ¼arc sin ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

εr2xðεr2z−εr1Þ

εr1ðεr2z−εr2xÞ

s:(11)

Notice that the Brewster angle for s-polarized waves is

also expected if we consider biaxial anisotropy where

ε2x≠ε2y≠ε2z. These results open new avenues in controlling

the polarization of waves by exploiting temporal interfaces. For

instance, Xu et al.45 reported complete polarization conversion

using anisotropic temporal slabs.

2.4 Temporal Switching in the Presence of Material

Dispersion

All previous results have assumed that the materials involved

are nondispersive, 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 non-

local response (dispersion) of a material can be considered

by assuming a susceptibility kernel in the form DðtÞ¼

ε0½ε∞EðtÞþRdt0χðt; t0ÞEðt−t0Þ,whereε∞is the background

relative permittivity and χðt; t0Þis the time-dependent electric

susceptibility, which must satisfy causality. Recently, the

generalization of the Kramers–Kronig relations has been intro-

duced for both adiabatic46 and nonadiabatic47 time-varying

susceptibilities χ. Research on temporal switching with

material dispersion dates back to Fante’sandFelson’swork

in the 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.48–51

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 Ebecomes 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 P:

d2P

dt2þγdP

dtþω2

0P¼ε0ω2

pE;(12)

where γis the collision damping rate, ω0is the natural frequency

of the free electron gas, and ωp¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Ne2∕ðmeε0Þ

pwith the volu-

metric carrier density N, and the electron’s mass me, and charge

e. A time-switched Nwas considered in Refs. 29 and 30 to ef-

fectively change ωpabruptly. The boundary conditions at the

temporal interface turn out to be the continuity of B,D,E

(and therefore P) and dP∕dt. The additional degrees of freedom

thereby introduced can give rise to a total of four (distinct or

degenerate) solutions for the new eigenfrequencies ω1;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, the experimental work demonstrat-

ing 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.52 In addition to lim-

itations in the achievable modulation speed and strength, one

chief difficulty also lies in altering a medium in its entirety

to conserve momentum in all directions. Huge energy ex-

changes and stringent simultaneity of switching are typically

required. Instead, more and more effort has been geared toward

switching materials only for spatially finite structures or in

lower dimensions. For instance, temporal reflections were

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-6 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

studied in Refs. 53 to 55 for two-dimensional (2D) graphene

plasmons, as shown in Fig. 4(a). In one-dimensional (1D) trans-

mission lines, broadband and efficient impedance matching can

be achieved by time-switching, as shown in Fig. 4 from Ref. 56.

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 that support resonances, we can extend temporal

switching to other applications. Switching resonant cavities

enables possibilities for plasma radiation61 and photon

generation.62 Similar ideas have also been applied to induce

static-to-dynamic field radiation in a switched dielectric brick

sandwiched in a waveguide.63 Li and Alù explored a time-

switched Dallenbach screen, whose schematic is shown in

Fig. 4(c), showing that time-switching can extend the absorption

bandwidth by changing the permittivity of an absorber while a

broadband signal enters it.57 On a related note, Mazor et al. un-

veiled how an excitation can be unitarily transferred between

coupled cavities by switching the coupling strength, even if

the detuning between the cavities is large.58 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 ad-

mittance, the energy stored in the LC resonator on the left can be

transferred to the one on the right efficiently, with robustness

to large detuning between the two resonance frequencies.

Subwavelength time-modulated meta-atoms have also been ex-

plored by Tretyakov’s and Fleury’s groups,64 developing a con-

sistent description of the interplay between time-modulation and

dispersive polarizability,65,66 whereas Engheta’s group recently

extended this concept to anisotropic meta-atoms, using it to

demonstrate temporal wave deflection,59 as shown in Fig. 4(e).

Finally, abrupt changes to the structural dispersion of a cav-

ity, realized, e.g., by switching part of a structure, constitute yet

another intriguing degree of freedom to be exploited. For in-

stance, in Ref. 60, Miyamaru et al. achieved ultrafast terahertz

frequency shifts by metalizing the upper boundary of a single-

metalized waveguide at subpicosecond timescales via photocar-

rier excitation, thereby modifying its dispersion relation and

inducing a large frequency shift, as shown in Figs. 4(f) and

4(g). Similarly, nonlinear terahertz generation was reported later

in ultrafast time-varying metasurfaces.67 Abruptly altering a spa-

tial boundary can be interpreted heuristically as switching the

effective permittivity of the waveguide structure,68 which also

enforces a temporal interface on the fields inside the waveguide.

To summarize, in this section, we discussed fundamental

aspects of the electromagnetics of time interfaces: we revisited

the temporal boundary conditions for Maxwell’s equations and

highlighted a few representative works on temporal scattering

and interference. Further theoretical opportunities include the

Fig. 4 (a) Temporal reflection of graphene plasmons reported in Ref. 54. (b) Impedance matching

using a time-switched transmission line.56 (c) Time-switched thin absorber in Ref. 57. (d) Unitary

excitation transfer between two coupled circuit resonators.58 (e) Temporal deflection caused by

isotropic-to-anisotropic switching on meta-atoms.59 (f) Temporal switching of structural dispersion

for ultrafast frequency-shifts, proposed in Ref. 60. (g) Corresponding frequency spectra for the

switching in (f). Figures adapted from Refs. 54,56,57,58,59, and 60.

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-7 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

study of more sophisticated material models, such as non-

Hermitian, anisotropic, and even bianisotropic switching mech-

anisms, 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 experi-

ments, some of which we will discuss in Sec. 5. In the next sec-

tion, we make use of the building blocks of temporal switching

discussed here to construct (quasi-)periodic temporal structures

and investigate some of the numerous opportunities that stem

from more complex time-varying systems.

3 Photonic Time-Crystals

The simplest mechanical archetype of a periodic time-

modulated wave system is a pendulum, whose length is peri-

odically modulated in time, one historic example being the

“Botafumeiro,”an 80 kg incensory at the Cathedral of

Santiago de Compostela, whose chain is periodically pulled

up and down via a pulley by monks, reaching speeds of

68 km∕hover arc lengths of 65 m within 17 modulation

cycles69 (a more familiar one is the child swing, whereby the

height of the center of mass is modulated). In fact, a character-

istic feature of this gain effect (called 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. 2to construct the building blocks of parametric amplifica-

tion in electromagnetism. After a first time-switching leading

to a change of impedance Z1¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

μ1∕ε1

p→Z2¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

μ2∕ε2

the displacement field at time twill comprise forward and back-

ward waves according to: D2ðtÞ∝T12ejω2tþR12 e−jω1t.Ifwe

now switch the material again after a time Δt2, 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 Δt3:

D2∕D0∝ðT12ejω2Δt2T23 þR12 e−jω2Δt2R23Þejω3Δt3

þðT12 ejω2Δt2R23 þR12e−jω2Δt2T23Þe−jω3Δt3:(13)

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 Ri;iþ1, respectively.

To study in more detail the effect of a periodic modulation,

let us set the final parameters in a double-switching process

equal to the initial ones, ε3¼ε1and μ3¼μ1, which forms

one modulation cycle. Evaluating the energy content of the

forward and backward waves, we arrive at

jT2j2¼1þ1

2ðZ2

1−Z2

2Þ2

1Z2

sin2ðω2Δt2Þ;(14)

jR2j2¼1

2ðZ2

1−Z2

2Þ2

1Z2

sin2ðω2Δt2Þ;(15)

so that it is evident that the relation: jT2j2¼1þjR2j2must al-

ways hold true regardless of the extent and duration of the “tem-

poral slab.”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 E×Hor the Minkowski (canonical)

momentum, D×Bmust 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

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 the power

in the forward-scattered waves can only be reduced by a scatter-

ing process. Finally, it is worth pointing out 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).

With these dualities in mind, we are now ready to study the

key features of infinite time-periodic (and quasiperiodic) systems,

which we refer to here as PTCs, although it should be noted that

this term bears a slight ambiguity with the concept of time-crystal

in condensed matter physics,70,71 where it denotes a stable phase

of matter that spontaneously breaks continuous time-translation

symmetry. Here, we intend PTCs as active systems relying on an

externally induced modulation of their constitutive parameters,

and whose underlying pumping mechanism is unaffected by

the linear wave dynamics imparted onto a probe wave.

In a conventional photonic space-crystal (PSC), it is well

known that Bragg scattering between waves separated in mo-

mentum by integer multiples of a reciprocal lattice vector g

of the crystal leads to the formation of bandgaps in energy,

in correspondence with the high-symmetry points in reciprocal

space (in 1D, this would happen at k¼0;g∕2). In these gaps,

the states have an imaginary momentum component, thus

decaying into the crystal (growing states are forbidden by en-

ergy conservation). It is expected, therefore, that modulating

a system periodically in time at frequency Ωwill yield bandgaps

in momentum (k-gaps) near the frequency Ω∕2, where Ωplays

the role of a reciprocal lattice vector, as a result of the inter-

ference between the waves that are forward-scattered by the

modulation, and those that are time-reversed. However, while

in a PSC, the wavevector is imaginary inside the bandgap; in

a PTC it is the energy that acquires an imaginary component

within a k-gap. As expected from our previous discussion on

momentum and energy conservation, while waves must spa-

tially decay into the PSC to preserve energy, they can only grow

in time as they interact with a PTC, to preserve momentum. This

duality is shown in Figs. 5(a) and 5(b). Let us consider an in-

finite series of switching events between ε1and ε2, as shown for

a few cycles in Fig. 5(b). Since the problem is periodic, the

Floquet (Bloch) theorem holds, so the solution may be written

as ψðtÞ¼eiΩtϕðtÞ, where ϕðtÞ¼ϕðtþTÞand T¼2π∕Ωis

the period of the modulation. For the specific case of a step-

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-8 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

like modulation, the temporal scattering coefficients can be re-

lated to each other by exploiting the symmetry T12 þR12 ¼

T21 þR21 ¼1, so the Floquet bands [shown in Fig. 5(c)] can

be calculated analytically.72 Notice how the k-gaps hosting

the amplifying states open at ω¼Ω∕2(i.e., ωT¼π, where

Tis the modulation period). The reason for this can be read

off the solution for the double-switching problem in Eq. (14–

15). Notice how the time-interval Δt2between the two switch-

ing events determines whether or not gain will occur: if this

interval, which corresponds to half of a modulation period T,

is a quarter of the wave period (i.e. Δt2¼π∕ð2ω2Þ), so that

Ω¼2π∕T¼2ω2, then at each switching process a net energy

input from the contribution of the sinusoidal term in Eq. (14)

will be coherently fed into the waves, leading to exponential

growth or, in the presence of dominant losses, loss-compensa-

tion.73 Related ideas were recently investigated, such as “multi-

layer”temporal slabs to design temporal transfer functions, in

analogy with multilayer matching filters in spatial transmission

line theory,74,75 while the effects of modulating both dielectric

and magnetic parameters in a PTC were studied in Ref. 76.

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 sym-

metry-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 re-

cently been answered in the affirmative, although experimental

observations of temporal topological edge states are still

missing.72 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(d).72

Far from a trivial generalization, 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 specifi-

cally later than) a specific instant of time, at which the proper-

ties 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 the

underlying wave dynamics is parametric amplification, and

the transient wave that constitutes the edge state has the effect

of temporarily opposing the exponential growth, as shown in

Figs. 5(e) and 5(f).72

In spite of time being a 1D quantity, time-modulation can

also provide a pathway toward higher-dimensional topological

effects. This can be accomplished by constructing synthetic

frequency dimensions, 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 shown in

Figs. 6(a)–6(c). Similarly to electrons in a simple 1D lattice,

photons in, e.g., a periodically modulated ring resonator may

be described by a tight-binding Hamiltonian

Fig. 5 (a), (b) Example of a (finite) layered (a) PSC and (b) PTC, and corresponding wave behav-

iour within the respective band gaps. (c) Band structure of a PTC, showing k-gaps and Zak

numbers of the different bands. (d) Change of Zak phase across the first k-gap. (e) Example of

an edge between two PTCs with opposite Zak phase. (f) Numerical demonstration of the effect of

a temporal edge between two topologically inequivalent PTCs: the parametric amplification pro-

cess is interrupted and the wave amplitude is depleted for a few periods following the temporal

edge, forming a temporally localized state. Figures adapted from Ref. 72.

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-9 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

H1¼gX

mða†

mþ1ameiϕþa†

mamþ1e−iϕÞ;(16)

where a†

mand amare the photon creation and annihilation

operators for the m’th mode on the lattice, g¼κΩ∕2πis the

hopping amplitude, κis the modulation strength, Ωis the

modulation frequency, and ϕis the modulation phase.77 In

fact, just as electrons hop between sites of a lattice, a dynami-

cal modulation can make photons hop between different

modes of the resonator, as long as the modulation frequency

is similar to the difference ΩR¼2πvg∕Lbetween the modes

of the structure, where vgand Lare the group velocity of the

waveguide constituting the resonator and Lis its length. The

simplest instance of such a lattice may be realized with a sin-

gle ring resonator combined with a periodic modulator, shown

schematicallyinFig.6.

Importantly, such a ladder does not need to be unidimen-

sional; 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 fre-

quency dimension, between modes,79 which means that a linear

lattice can now be equivalently described as a folded one to

highlight its higher dimensionality, as shown in Figs. 6(f)–6(h).

With time-modulation, this can be done by considering two

periodic modulations of different frequencies, e.g., Ω1¼ΩR

and ΩN¼NΩR, thus introducing coupling between N’th-

neighboring modes in the lattice, so the Hamiltonian above

can be generalized to

HN¼gX

mða†

mþ1ameiϕþh:c:ÞþgNX

mða†

mþNameiϕNþh:c:Þ;

(17)

where gN¼κNΩR∕2π. Here, the second term accounts for the

coupling between modes separated by a frequency gap NΩR,

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 momentum80 or

spatial lattices79 to implement a range of photonic effects, from

nontrivial Chern numbers to effective gauge fields and spin-

momentum locking. Implementations of synthetic frequency

dimensions have been realized in RF with varactors81 and

superconducting resonators,82 and in the near-infrared with

fiber and on-chip electro-optic modulators,83,84 as well as all-

optically via four-wave mixing, with frequency spacings gen-

erally ranging from MHz to THz.77 A recent proposal makes

use of a single ring-resonator coupled to an array of optome-

chanical pumps and a single electro-optic modulator to realize

an on-chip 2D topological insulator with two synthetic

dimensions.85

Fig. 6 (a) The resonant modes of a structure form a lattice in the frequency dimension.

(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 po-

tential. (d) A ring resonator combined with a modulator can reproduce such a model, (e) enabling

hopping between its equally spaced resonant modes.77 (f)–(h) A synthetic dimension may be con-

structed 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.78

(i), (j) A spatially 1D chain of ring resonators can be combined with temporal modulation to form

a synthetic 2D lattice.79 Figures adapted from Refs. 77–79.

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-10 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

3.2 Non-Hermitian and Disordered Photonic

Time-Crystals

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 ampli-

fication in the form of a negative dissipation rate. However,

as discussed above, time-modulation offers an alternative

route toward gain. 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-modula-

tion cannot provide loss, as a consequence of momentum con-

servation, whereas an imaginary component in the response

parameters of a medium can be used to provide either gain or

dissipation depending on its sign.

This distinction between parametric and non-Hermitian gain

makes it natural to ask what more can be achieved by modulat-

ing in time the non-Hermitian component of the material re-

sponse. As discussed in Sec. 2, the switching of dissipation

not only impacts the amplitude of a wave but also its phase

and can also generate backward waves, leading to highly non-

trivial temporal wave scattering. In one instance, it was recently

shown that non-Hermitian time-modulation can be used for

nonreciprocal mode-steering and gain in frequency space.86

Consider the setup shown in Fig. 7(a): two resonators with

mode-lifetimes γ1and γ2are coupled via a time-dependent cou-

pling constant k0þΔk0cosðΩtÞþjΔk00 sinðΩtÞ, 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 non-

reciprocal gain.86

The link between time-modulation and PT symmetric

Hamiltonians has also been discussed in some depth for the

parametric resonance case, where the frequency of the modula-

tion doubles that of the incoming light.88 Elegantly, PT

Fig. 7 (a) Two resonators coupled via a combination of constant (k0), time-dependent Hermitian

(Δk0), and time-dependent non-Hermitian (Δk00) coupling terms. (b)–(e) Wave amplitude under

excitation via port 1 in the two resonators (orange and blue) for two arbitrary values of Δk0at

the excitation frequency (ω1) and the two sidebands for (b) k0≠0 and Δk00 ¼0 (energy stored

in both sidebands and both resonators); (c) Δk00 ¼0 and k0¼0 (energy channeled to sideband

frequencies is all in the second resonator while the first only hosts the input frequency);

(d) Δk0¼Δk00 and k0≠0 (energy gained is nonreciprocally channeled into the upper sideband

and it is distributed over both resonators); (e) Δk0¼Δk00 and k0¼0 (energy gained is nonreci-

procally channeled only into the upper sideband and uniquely in the second resonator).86

(f) A random sequence of temporal δ-kicks results in (g) universal statistics observed by the energy

in the system U. The energy at the N’th kick UNis plotted against the step number N.87 Figures

adapted from Refs. 86 and 87.

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-11 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

symmetry breaking in these parametric systems can be related

to the stability conditions of the Mathieu equation governing a

harmonically modulated structure.89 Such time-modulation-

induced PT symmetric physics additionally enables not only

lasing instabilities but also bidirectional invisibility due to aniso-

tropic transmission resonance, when two PT-symmetric time-

Floquet slabs are combined together such that their respective

parametric oscillations cancel out exactly. New perspectives re-

lating exceptional point physics and parametric modulation may

be also found in Refs. 90 to 92, whereas Refs. 93 and 94 inves-

tigate topological phases in non-Hermitian Floquet photonic

systems.

Another interesting direction for non-Hermitian time-modu-

lated systems relates to the temporal analog of the causality

relations of conventional passive media.46 As a well-known con-

sequence 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 function of a

system must relate to each other through the Kramers–Kronig

relations. However, the spectral analog 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-modu-

lation 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 fre-

quency dimensions.46,95

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.87,96–98 Interestingly, however, the conven-

tional localization observed in spatially disordered media does

not occur in a time-modulated system. Instead, in a similar fash-

ion to the occurrence of edge states, the onset of the temporal

analogue of Anderson localization manifests itself as an expo-

nential growth in the energy of the waves. Such wave dynamics

has been investigated both analytically,87 demonstrating the uni-

versal statistics occurring in δ-kicked temporal media, and nu-

merically,97 as a disordered perturbation in a PTC. Experiments

on wave dynamics in temporally disordered systems have also

recently been performed with water waves.96

In the following section, we move on to briefly discuss

temporally periodic surfaces, whereby the interplay of spatial

discontinuities and temporal modulation can enable not only

several realistic platforms for implementations with metasurfa-

ces (which we delve further into in Secs. 4.4 and 5) but also

completely new regimes of wave scattering.

3.3 Time-Modulated Surfaces

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 oc-

cupied 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 (kx) and out-of-plane (kz)momentum

via both its in-plane structure and its transverse boundary,

such that they must add in quadrature to the total momentum

k0¼ﬃﬃﬃ

pω∕c0of the impinging waves. Interestingly, time-

modulated surfaces enable sharply different wave dynamics,

due to the fact that the light cone treats the frequency axis

on a different footing compared to the momentum axes; if

we consider light impinging on a flat, time-modulated surface,

its in-plane momentum kxwill now be conserved, whereas kz

and, importantly, the wave frequency ωare now allowed to

change. Taking a vertical cut of the light cone for a fixed value

of kx, we see that the two quantities being traded by the sur-

face, kzand ω, 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 such as an SNOM tip99 or periodically using a

grating that induces a Wood anomaly.100 However, the argument

above can be exploited to envision surface-wave excitation via a

temporal grating,101 as shown in Figs. 8(d)–8(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 polari-

tonic 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 frequency conversion.103 The advent of metasurfaces

across the electromagnetic spectrum has led to a surge of oppor-

tunities for advancing this known field of research both

theoretically104–112 and experimentally.102,113–116 Recent imple-

mentations of frequency translation with metasurfaces have

been recently carried out in the microwave regime,102 and im-

plementations with experimentally more practical discretized ar-

rays of time- and space–time-modulated reactive elements are in

steady development.107,114 Several more experimental implemen-

tations are discussed in Sec. 5. Interestingly, the combination

of bianisotropy and time-modulation has also been predicted

to give rise to a nonreciprocal response (see Sec. 4.1).117

Specific scattering studies have also been dedicated to periodi-

cally modulated slabs,118 particularly in the context of paramet-

ric amplification.112

This recent surge of interest in time-modulated media has

certainly highlighted the fundamental nature of the new wave

phenomena that time-modulation can unlock, and it is evident

that a wealth of research opportunities with PTCs, such as their

implications in quantum systems and finite-size structures, still

lies largely unexplored. Further directions involving space–time-

modulated surfaces have nevertheless attracted remarkable inter-

est already (Sec. 4.4). However, before discussing them, it is

instructive to extend the theoretical background presented so far

to the case of space–time modulations, which introduce a wealth

of additional scattering phenomenology. At the end of Sec. 4,

we combine these ideas on modulated surfaces with those of

space–time media to study space–time metasurfaces (Sec. 4.4).

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-12 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

4 Spatiotemporal Metamaterials

Spatiotemporal modulations generally imprint a traveling-wave-

like perturbation, of the form δεðx; tÞ¼δεðgx −ΩtÞ(and sim-

ilarly for μ) onto the response parameters of a medium, where

g¼2π∕Land Ω¼2π∕Tare the spatial and temporal compo-

nents of the reciprocal lattice vector associated with the spatial

and temporal periods Land Tof the modulation, which form a

spatiotemporal lattice vector p¼ðT; LÞ, as shown in Figs. 9(a)

and 9(b).119 The idea of traveling-wave modulation has been in

use since the early days of traveling-wave amplifiers, whereby

electron beams are used to pump energy into copropagating

microwave beams122 and underwent a number of investigations

in the 1960s123–127 and later in the early 2000s.128,129 Recently,

the interest in these systems has revamped, with intense research

efforts in the modeling and realization of space–time-modulated

nonreciprocal systems (see Sec. 4.1).25,121,130–134 In space–time me-

dia, neither frequency nor momentum is individually conserved,

but rather a spatiotemporal Bloch vector ðω;kÞforms a good

quantum number, and the fields can generally be expressed as

a superposition of a discrete set of Floquet–Bloch modes charac-

terized by frequency-momentum vectors ðkþng; ωþnΩÞ,

where n∈Zas shown in Figs. 9(c)–9(f), so an eigenmode will

have the form

ψðx; tÞ¼ejðωt−kxÞX

anejnðΩt−gxÞ;(18)

andaneigenvalueproblemmaybesetintermsofeitherωðkÞor

kðωÞ. Importantly, these new reciprocal lattice vectors ðg; ΩÞare

not generally horizontal as in a PSC [Fig. 9(c)], nor vertical as in a

PTC [Fig. 9(f)], but may form an arbitrary angle in the ω−k

space, which may be smaller than the slope c0of the bands for

the background medium (subluminal) [Fig. 9(d)] or larger (super-

luminal) [Fig. 9(e)], reducing to a pure temporal modulation in the

limit where the modulation speed vm¼Ω∕g→∞as the entire

medium is then effectively modulated at the same time [Fig. 9(f)].

Interestingly, a peculiar regime of modulation velocities:

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1þmaxðδεÞ

p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1þmaxðδμÞ

p<v

<c0

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1−jminðδεÞj

p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1−jminðδμÞj

p;(19)

exists, where the Fourier expansion in Eq. (18) does not con-

verge, as first pointed out by Cassedy in Ref. 135. We discuss

this exotic luminal regime120 in detail in Sec. 4.3. In the next

section, we discuss the role of space–time modulation for the

Fig. 8 (a) Light scattered from a flat, time-varying metasurface can access a locus of states

that 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.102 (d) Illustration of surface-

wave excitation via the Wood anomalies using (e) spatial and (f) temporal modulation of (g) a

surface. (h) Example of the Wood anomaly observed in transmission, the surface-wave excitation

is explicitly shown in (i) from a finite-element-time-domain simulation.101 Figures adapted from

Refs. 102 and 101.

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-13 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

engineering of nonreciprocal scatterers, which first motivated

the interest in this field.

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 earned a spotlight in 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.121 Nonreciprocal components such as isolators and

circulators are crucial for duplex communications, enabling si-

multaneous transmission and reception at the same frequency on

the same channel without parasitic reflections, as well as other

applications such as laser protection from backscattering and

field enhancement. The common approach to breaking this sym-

metry 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 electro-

magnetism 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.121

The basic idea of using ST modulation for optical isolation

may be summarized in Fig. 9(g); the modulation produces tran-

sitions between two optical states that differ in both momentum

kand frequency ω, 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 ω1may be com-

pletely converted into a new mode with frequency ω2, a back-

ward wave at ω1will not be converted as it will be mismatched

in frequency and/or momentum. Hence, by introducing a nar-

rowband filter at ω2, the system allows propagation only in

the backward direction, thus achieving optical isolation,121,132

in a similar fashion to the way a diode conducts electricity

unidirectionally.136 An equivalent way of viewing this scattering

asymmetry in an extended medium is the opening of an asym-

metric bandgap in the photonic dispersion of a material,

allowing propagation along one direction only.137

Ring-resonators constitute perhaps the most promising com-

ponents 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 Fig. 10(d). Due to the split-

ting 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 res-

onator 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-res-

onance, thus coupling poorly to the mode of the resonator and

being largely transmitted to the bottom port of the channel

waveguide.138 This strategy can be used to realize electromag-

netic circulators, as shown in Figs. 10(f)–10(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

Fig. 9 (a) Space–time diagram of a space–time crystal with alternating refractive indices niand nj,

spatiotemporal lattice vector pand velocity vm. (b) Double-period space–time crystal, character-

ized by two different lattice vectors pAand pB, and two different velocities vm1and vm2.119 (c) The

band structure for a spatial crystal with ε∼cosðgz −ΩtÞbecomes asymmetric as (d) a finite tem-

poral component Ω, and therefore a finite modulation velocity vm¼Ω∕g, are 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 [(c), (d)] a

subluminal and [(e), (f)] a superluminal regime.120 (g) Nonreciprocal mode-coupling in a space–

time-modulated waveguide: the modulation can couple forward modes between them 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.121 Figures adapted

from Refs. 119,120, and 121.

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-14 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

impeding transmission in the opposite sense (i.e., 2→1,3→2,

and 1→3). Figure 10(f) shows the transmission from channels

1 to 2 and 1 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)].139 Other electro-

magnetic implementations have been realized with silicon

waveguides,133 as well as microstrip transmission lines,130 with

modulation frequencies ranging from hundreds of MHz to tens

of GHz.

Multiple extensive reviews on space–time modulation focus-

ing 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.121,140

4.2 Synthetic Motion with Space–time 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.18,141–143 Moreover, the electromag-

netic response of a moving system is inherently nonreciprocal,

as it is associated with a broken time-reversal symmetry.144–146 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,146 classical and quantum

non-contact friction,147–150 and parametric amplification146,151–153

among others.3

Fig. 10 (a)–(c) Angular momentum biasing: a directional modulation along a ring resonator splits

the degeneracy between clockwise and anticlockwise traveling states (a),138 in analogy with the

Zeeman splitting of electronic states in a magnetic field (b),138 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 (c).139 (d) Example of operation of a nonreciprocal 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 (e) the direction of excitation

determines the efficiency of the coupling to the resonator, and hence of the transmission to

the output port of the channel waveguide.138 (f), (g) Theoretical and (h) experimental performance

of an RF circulator made with angular-momentum biased resonators: (f) and (g) the reciprocal

response in the absence (f) and presence (g) of angular biasing, and (h) the performance of

the experimental implementation of the circulator.139 Figures adapted from Refs. 138 and 139.

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-15 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

Unfortunately, it is impractical to take technological advan-

tage of many of the features highlighted in the previous para-

graph, 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 .149,150,154–158 For example, it was recently

experimentally verified that drifting electrons in graphene can

lead to nonreciprocity at terahertz frequencies and to a Fresnel

drag.159,160 A different way to realize an effective moving re-

sponse without any actual physical motion is through a traveling

wave space–time modulation.119,124,126,135,161,162 Such a solution is

discussed next in detail.

A system featuring a traveling-wave-type modulation is char-

acterized, in the dispersionless limit, by the following constit-

utive relations:

D

B¼εðx−vmt; y; zÞ10

0μðx−vmt; y; zÞ1·E

H;

(20)

where vmis the modulation speed [Fig. 11(a)]. This form of

modulation in space and time imparts a synthetic motion to

the material response along the xdirection with a uniform

speed vm. 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

x0¼x−vmt; y0¼y; z0¼z; t0¼t; (21)

preserves the usual structure of Maxwell’s equations, with the

transformed fields related to the original fields as follows

D0¼D,B0¼B,E0¼Eþvm×B,H0¼H−vm×D, and

vm¼vm

x. In the new coordinates, the constitutive relations

are time-independent

D0

B0¼ˆ

ε0ðr0Þ

ξ0ðr0Þ

ζ0ðr0Þˆμ0ðr0Þ·E0

H0;(22)

with r0¼r−vmtand the transformed effective parameters

given as

ε0¼ε1

1−εμv2

1tþˆ

x⊗ˆ

x;(23)

ˆμ0¼μ1

1−εμv2

1tþˆ

x⊗ˆ

x;(24)

Fig. 11 (a)–(c) A synthetic space–time-modulated crystal (a) is formally equivalent to a fictitious

moving bianisotropic time-invariant crystal (b) when vm<c0. The velocity of the equivalent mov-

ing crystal is identical to the modulation speed vm; in the long-wavelength limit, the system re-

sponse can be homogenized (c) and the crystal behaves as a uniaxial dielectric moving with

speed vD. The sign of vDis not necessarily coincident with the sign of v. (d) Dispersion of waves

in the effective medium as shown in (c). When only one of the material parameters is spatiotem-

porally modulated, the dispersion is symmetric, and the equivalent drag velocity vanishes (blue

lines). However, when both εand μare modulated, the dispersion is asymmetric (red lines, as-

suming vD>0). The waves that propagate in the direction of vDhave a larger group velocity than

the waves that propagate in the opposite direction.161 (e) Drag velocity vDand (f) effective bia-

nisotropic coefficient ξef as a function of the modulation speed vmfor a representative space–

time-modulated crystal: the velocity of the equivalent moving medium and the magnetoelectric

coupling coefficient flip sign in the transition from the subluminal to the superluminal regime.

The luminal regime is marked as a shaded area in (e) and (f). Figures adapted from Ref. 163.

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-16 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

ξ0¼−

ζ0¼−εμ

1−εμv2

vm×1:(25)

In the above, 1t¼ˆ

y⊗ˆ

yþˆ

z⊗ˆ

zand ⊗represents the ten-

sor product of two vectors. As seen, the coordinate transforma-

tion originates a bianisotropic-type coupling determined by

the magnetoelectric tensors

ξ0and

ζ0,164 such that the electric

displacement vector D0and the magnetic induction B0depend

on both the electric and magnetic fields E0and H0. Indeed, as

further discussed below, one of the peculiar features of the trav-

eling-wave space–time modulation is that it mixes the electric

and magnetic responses, leading to the possibility of a giant

bianisotropy in the quasistatic limit.161,163

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 [Eq. (21)]. Usually, this does not create any dif-

ficulties, but it is relevant to mention that it is also possible to

obtain a time-invariant response with a Lorentz coordinate trans-

formation. Such a solution is restricted to subluminal modula-

tion velocities, and thereby the Galilean transformation is

typically the preferred one.

Interestingly, the constitutive relations [Eq. (22)] in the new

coordinate system are reminiscent of those of a moving dielec-

tric medium.144 Due to this feature, the electrodynamics of

space–time media with traveling wave modulations bears many

similarities to the electrodynamics of moving bodies.161 Yet, it is

important to underline that a space–time-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

vmis nontrival, very different from the constitutive relations

[Eq. (20)]. Below, we revisit this discussion in the context of

effective medium theory.

Even though the space–time-modulated dielectric system is

not equivalent to a moving dielectric, its response can be pre-

cisely linked to that of a fictitious moving medium in the sub-

luminal case. In fact, as previously noted, a suitable Lorentz

transformation makes the response [Eq. (20)] of a space–time

system independent of time, analogous to Eq. (22). Thus, the

original constitutive relations Eq. (20) are indistinguishable

from those of a hypothetical moving system with a bianisotropic

response in the co-moving frame of the type [Eq. (22)]. In other

words, the synthetic motion provided by the traveling 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 traveling wave modulation. In the following, we

focus our attention on periodic systems and in the long wave-

length regime. Effective medium methods have long been used

to provide a simplified description of the wave propagation in

complex media and metamaterials.165–168 The effective medium

formalism is useful not only because it enables analytical mod-

eling of the relevant phenomena but also because it clearly pin-

points the key features that originate the peculiar physics of

a system.

The standard homogenization approach is largely rooted in

the idea of “spatial averaging,”so the effective response de-

scribes 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 simul-

taneously 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 traveling wave mod-

ulations can be homogenized using standard ideas.163,169,170

In fact, since a traveling wave modulated spacetime crystal is

virtually equivalent to a moving system, it is possible to find

the effective response with well-established methods by work-

ing in the co-moving frame (primed coordinates) where the

medium response is time-invariant.163

To illustrate these ideas, we consider a 1D photonic crystal

such that the permittivity εand permeability μare independent

of the yand zcoordinates. It is well known that, for stratified

systems, the components of the Eand Hfields parallel to the

interfaces (yand zcomponents) and the components of the D

and Bfields normal to the interfaces (xcomponents) are con-

stant in the long-wavelength limit,171 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

hD0iand hB0ito the spatially averaged hE0iand hH0i.163 The

effective parameters in the original (laboratory) frame can then

be found with an inverse Galilean (or Lorentz) transformation.

Such a procedure leads to the following constitutive relations in

the lab frame:

hDi

hBi¼ˆ

εef ξef

x×1

−ξef

x×1ˆμef ·hEi

hHi;(26)

where ˆ

εef ,ˆμef , and ξef are some parameters that can be written

explicitly in terms of the permittivity and permeability profiles

εðxÞand μðxÞ.163 It turns out that when only one of the material

parameters is modulated in spacetime the magnetoelectric cou-

pling coefficient ξef vanishes and the effective medium behaves

as a standard uniaxial dielectric.161,163 For example, if the

material permeability is independent of space and time, then

ξef ¼0and ˆμef ¼μ1, for any permittivity profile εðxÞ. This

property can be intuitively understood by noting that when

μis a constant, the “microscopic”constitutive relation

B¼μHimplies that the averaged fields are also linked by

hBi¼μhHi, 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.166,172

Remarkably, when both εand μare modulated in spacetime,

ξef 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 bianiso-

tropic 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,164 having a

nontrivial bianisotropic response in the long wavelength limit is

a rather unique result. (For completeness, we point out that some

antiferromagnets, such as chromium oxide, may be character-

ized by an intrinsic axion-type bianisotropic response in the

static limit.)173,174 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 conven-

tional metamaterial are necessarily decoupled in the static limit

and there is no bianisotropy. In contrast, for a space–time-modu-

lated system, the electric and magnetic fields are never fully

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-17 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

decoupled, as the time modulation of the material implies al-

ways some nontrivial time dynamics. The strength and sign

of ξef can be tuned by changing the modulation speed and

the profiles of the permittivity and permeability. As shown in

Fig. 11(f),ξef 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.175 The transition from the sublumi-

nal to the superluminal regime is marked by a change in sign of

ξef . Furthermore, it can be shown that constitutive relations

[Eq. (26)] are identical to those of a fictitious moving aniso-

tropic dielectric characterized in the respective co-moving frame

by the permittivity ˆ

εDand by the permeability ˆμD. The equiv-

alent medium moves with a speed vDwith respect to the lab

frame. The parameters ˆ

εD,ˆμD, and vDare determined univocally

by ˆ

εef,ˆμef , and ξef .163 In general, vDmay be rather different from

the modulation speed vm, both in amplitude and sign. The veloc-

ity vDcan be nonzero only if εand μare both modulated in

spacetime, i.e., only if ξef ≠0. Therefore, it follows that a

space–time-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 c0[Fig. 11(e)]. Thus, space–

time-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 space–time-

modulated crystal can produce a drag effect.161 Specifically, a

wave copropagating with the equivalent moving medium, i.e.

toward the direction sgnðvDÞˆ

x, moves faster than a wave propa-

gating in the opposite direction, due to the synthetic Fresnel-

drag effect.143 A clear fingerprint of the Fresnel drag can be de-

tected in the dispersion of ωversus kof the electromagnetic

modes of the spacetime crystal. In fact, when both εand μ

are modulated in time, the slopes of the dispersion of the pho-

tonic states near k¼0depend on the direction of propagation of

the wave. The dispersion of the waves that copropagate with

the equivalent moving medium exhibits a larger slope than the

dispersion of the modes that propagate in the opposite direction,

as shown 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: ε∕μ¼const:. In fact, the dispersion of a matched pho-

tonic 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.

4.3 Luminal Amplification and Spatiotemporal

Localization

If the speed of a space–time modulation falls within the modu-

lation 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 un-

stable phase whose underlying mechanism is, however, com-

pletely distinct from the parametric amplification processes

described in Secs. 2and 3.120 In this regime, all forward bands

become almost degenerate, as the frequency-momentum recip-

rocal lattice vectors effectively align with the forward bands.

This regime marks a transition between the subluminal and

superluminal regimes in Figs. 9(d) and 9(e), respectively.

Due to the resulting strong degeneracy between forward-travel-

ing waves, the eigenmodes of these luminal systems cannot be

written in the Bloch form:176 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 Figs. 12(a)–12(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

trapping of field lines by the synthetically moving grating.178

The wave dynamics in a luminal medium can be captured by

the approximate equation for the energy density U:

∂ln U

∂t0¼vmþcl

2∂ln μ

∂Xþ∂ln ε

∂Xþðvm−clÞ∂ln U

∂X;

(27)

where vmis the grating velocity, clðXÞ¼ðεμÞ−1∕2and X¼x−

vmtand t0¼tare moving coordinates, which can be solved by

successive iterations.177 Note that this equation becomes exact if

ε¼μeverywhere (impedance-matching). The characteristic

feature of the luminal regime is the presence of points along

the grating where the local phase velocity clðXÞof the waves

matches the velocity vmof 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 in

Eq. (27), which acts as an energy source term, resulting from

the spatiotemporal change in εand μ. In fact, these local gra-

dients in the electromagnetic parameters 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 veloc-

ity-matching points determines whether these will function

as attractors [π∕2in Fig. 12(b)] or repellors [3π∕2in

Fig. 12(b)] as a result of the Poynting flux driving power toward

them or away from them.179 It has been noted that this com-

pletely linear mechanism results in pulse compression or

expansion.180 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,176 as shown in

Figs. 12(e)–12(g). Importantly, luminal amplification does not

rely on backscattering, as opposed to parametric amplification,

and in fact it occurs just as well in a system that is impedance-

matched everywhere in spacetime, in sharp contrast to paramet-

ric 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 mecha-

nism, e.g., a pump pulse, would generally need to impinge on

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-18 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

the system from an additional direction. In addition, field pen-

etration into a bulky structure is generally inhomogeneous, so in

practice only a small surface layer of a bulk medium would ef-

fectively be modulated. These considerations, combined with

the ease of fabrication of metasurfaces compared to bulk meta-

materials, have led to a concentration of research interest in

space–time metasurfaces, whereby a traveling-wave modulation

is applied to the surface impedance of a thin sheet, with propos-

als covering a wide range of ideas and applications, some of

which are shown in Fig. 10.

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 mag-

net-free nonreciprocity to surface implementations.104,111,182,183

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 the overall response of the structure can be

encoded as an array of bits (if only two values exist for each

element).184 Switching the individual impedances of a volt-

age-controlled array of such elements can enable exquisite

control over the metasurface 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)],114 as well as to perform photonic analog

computing.185 Furthermore, close to the luminal regime, oppor-

tunities for nonreciprocal hyperbolic propagation,162 as well as

vacuum

Cerenkov radiation186 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 strat-

egy consists of using a spatiotemporal modulation of a sur-

face 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 the (equally

spaced) frequencies of the outgoing waves, which emerge at

Fig. 12 (a) Time dependence of the electric field intensity at the output port transmitted through

different thicknesses (blue to red) dof luminal crystal. (b) In the gain region, the permittivity gra-

dient 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 (c) for input frequency ω0¼8Ω, with Ωthe modulation frequency, and (d) for a DC input

field (ω0¼0), showing the frequency generation responsible for the overall gain.120,177 (e),

(f) Trajectories of a point of constant phase along the spacetime variable X¼x−vmt: (e) outside

of the localization regime (resulting in oscillations that periodically compress and decompress the

pulse) and (f) within the localization regime, resulting in the amplification and compression

described in (a)–(d).176 Figures adapted from Refs. 120,176, and 177.

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-19 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

the same angle, as shown in Fig. 13(f).108 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.110,181 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 nonclassical light, with sev-

eral opportunities for the generation of entangled photonic

states, thereby anticipating a whole new scenery for this field

to expand toward.187

4.5 Further Directions

Among new directions for space–time media, dispersive effects

offer additional knobs for mode-engineering,188 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.189

Chiral versions of synthetically moving and amplifying electro-

magnetic media, mimicking the Archimedean screw for fluids,

have recently been proposed, and may be realized with circu-

larly polarized pump-probe experiments.190 Topology also occu-

pies a prime seat, with Floquet topological insulators having

been realized in acoustics,191 and theoretically proposed for

light.192 Further related opportunities have been demonstrated

for photonic realizations of the Aharonov–Bohm effect,193 as

well as for the engineering of Weyl points (three-dimensional

spectral degeneracies analogous to Dirac points, signatures of

topological states) within spatially 2D systems, exploiting a syn-

thetic frequency dimension.194 The idea of space–time media has

also been formalized in the interesting mathematical formu-

lation of “field patterns.”These objects arise from the consecu-

tive 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.170,195,196 Finally, besides photonic

systems, nonelectromagnetic dynamical metamaterials have

already acquired significant momentum, with several theoretical

proposals197–199 and experimental realizations, including asym-

metric charge diffusion,200 nonreciprocity,145,201,202 Floquet topo-

logical insulators191 and topological pumping203 among many

others. In particular, digitally controlled acoustic meta-atoms

have recently attracted significant interest to probe time-modu-

lation physics in acoustics.204,205

Having discussed the wealth of potential new physics en-

abled by temporally and spatiotemporally modulated media,

we now turn our attention to the latest technological advances

toward experimental implementations in photonics, and their

related challenges.

Fig. 13 Examples of applications of space–time metasurfaces. (a) An isolator based on the uni-

directional excitation of evanescent modes. (b) Waves scattering off an STM experience a non-

reciprocal phase shift. (c) An STM-based circulator.104 (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.114

(f) Illustration of wave power combining using STMs: incoming waves with frequency ω0coming

from a discrete range of different angles can be scattered toward the same outgoing direction by

imprinting the necessary frequency-wavevector shifts via the STM.108 (g) A proposal of an angular

momentum-biased metasurface for the generation of orbital angular momentum states: each azi-

muthal section is temporally modulated with a different relative phase, imparting the necessary

angular-momentum bias. Figures adapted from Refs. 104,108,114, and 181.

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-20 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

5 Experimental Advances and Challenges

in All-Optical Implementations

The experimental realization of time-varying effects constitutes

a new, broad, and fruitful field of research across the wider

wave-physics and engineering spectrum. Implementation in op-

tics and more particularly in nanoscale architectures and meta-

surfaces 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,206 magnetic,207 acousto-

optic,208 opto-mechanical,209 and electronic systems.133 In par-

ticular, electromagnetic metasurfaces can be modulated via

mechanical actuation, chemical reactions, or phase-change ma-

terials, as well as electrically.116 Nevertheless, their slow modu-

lation 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 realizations of time-

varying media in these optical frequency ranges. Emphasis is

placed on metasurfaces as these have subwavelength thick-

nesses that allow for purely temporal modulations of the

medium. In a metasurface, the various constraints of bulk media

such as absorption losses 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 first 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 ultra-

fast 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

Due 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, modu-

lation 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 to 10 fs regime). Optical modulation of a

medium stems from a change in ε, 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 semicon-

ductors 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 transi-

tions of the electrons within the active medium. The redistrib-

ution 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 εin a vari-

ety 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 ε, 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, whereas 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.210 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

aluminum zinc oxide (AZO) are well-known, silicon-technol-

ogy compatible TCOs with very strong nonlinear optical re-

sponse,211–214 which makes them good candidates as platforms

for time-varying metasurfaces. Particularly, intraband transi-

tions 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 2016,211 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

Δn∕n, where nis the refractive index and Δnis 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 Δn.

Time-refraction in pump probe experiments has been shown

in bulk ITO and AZO thin-films,215,216,219–221 with frequency shifts

as strong as 58 THz216 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

Figs. 14(c)–14(f). In the works of Ferrera et al.,219 Vezzoli

et al.,220 and Bruno et al.,216 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.220 This

demonstrated the potential of these TCOs for time-varying

applications, yet the strength of the modulation and signal

was mostly achieved due 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 an 80-nm-thick ITO meta-

surface was demonstrated in Liu et al.’s work222 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 signifi-

cantly smaller portions of space. In this work, it was shown

that, as expected, a shorter pulse will lead to a stronger shift

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-21 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

Fig. 14 (a), (b) Concept of photocarrier excitation and time-refraction in ITO:215 (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 experi-

ment at 1400 nm.216 (g) Strong coupling between a plasmonic antenna and an ENZ thin film:217

electric field jEj2distribution 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 depth.217

(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.218 The increase in modulating pump

intensities (0.5, 1, and 2 GW∕cm2) leads to a shift near zero delay. Figures adapted from

Refs. 216 to 218.

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-22 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

in frequency due to a larger dn∕dtduring the short propagation

time within the ITO slab, but more interestingly, that saturation

happens at lower energies for longer pulses. This indicates that

predicting and measuring the ultimate material response time for

short and intense modulation constitutes still an open research

question.

Nanostructuring ENZ materials is a powerful strategy to in-

crease the optical field enhancement and lead to efficient ultra-

fast modulation as demonstrated by Guo et al.’s work223 with

ITO nanorods. In this work, the thickness of the medium is

2.6μm, a length sufficient to cause significant loss upon propa-

gation. An alternative to nanostructuring the ITO to increase the

field enhancement consists of exploiting the plasmonic mode

exhibited by flat ENZ thin films.224 In Bohn et al.’s work,225

the plasmonic ENZ mode was excited using a Kretschmann con-

figuration, 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 ar-

chitectures to ensure efficient coupling, as we will address in the

next section.

5.1.2 Hybrid platforms for time-varying experiments

To compensate for the lack of interaction volume in a metasur-

face, good in-coupling and enhancement of the electric field is

necessary to achieve strong nonlinear optical modulation. ENZ

material-only metasurfaces lack strong resonant behavior, 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 cou-

pling and local field enhancement for otherwise impedance-

mismatched ENZ materials.

Plasmonic nanoantennas couple efficiently propagating radi-

ation 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 ef-

ficient 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 achieved217 [see

Figs. 14(g)–14(h)]. The coupled antenna-ENZ thin film enhan-

ces the field in the ENZ medium, and strong frequency shifts

originating from time-refraction, as well as efficient negative re-

fraction, have been measured from ITO films217,222,226 with a rec-

ord 11.2 THz frequency shift being recorded at a comparatively

low power of 4GW∕cm2in Pang et al.’s work218 [Figs. 14(i)–

14(k)]. Such time-varying metasurfaces are at the moment lim-

ited by the low damage threshold of the plasmonic antennas, the

damage threshold of ENZ materials such as ITO or AZO being

much higher. To circumvent damage threshold constraints, ar-

chitectures where the optical field is localized only in the ENZ

layer 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 superpo-

sition 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.,227 a layer

of In:CdO with an ENZ frequency at 2.1μmexhibited large

nonlinearities when illuminated at its Brewster angle, with

the ninth harmonic being generated and measured from the

metasurface. Time-varying effects are noticeable in the har-

monic spectrum, with, for example, the fifth harmonic exhibit-

ing two peaks: a small one at 5fand 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 improve-

ment in realizing time-varying metasurfaces using the favorable

properties of ENZ materials and other TCOs. This calls for fur-

ther investigation of the implementation of ENZ properties in

other types of metasurfaces, such as the multilayered systems

presented in Sec. 5.2.3.

5.1.3 High-index dielectric metasurfaces as time-varying

media

In the quest for resonantly enhanced materials for time-modu-

lation, high-index dielectrics provide an alternative to ENZ

structures, as they offer possibilities for various resonant archi-

tectures and a high damage threshold. In particular, nanoanten-

nas with Mie resonances or bound states in the continuum

allow for strong time-modulation by combining efficient cou-

pling to the active medium and a good field enhancement.

That is because even though Δn∕nis low in a high-index dielec-

tric, the change of phase of the metasurface can be strong when

the material is modulated near a resonance. Shifts in the reso-

nance of a photonic crystal228 were demonstrated, as well as in

Mie-resonant systems, in a range of materials ranging from

Si229–231 to GaAs,232 GaP,233 and Ge.234 GaAs exhibits stronger

modulation efficiencies as it features a direct bandgap in the vis-

ible to near-IR region, while Si, GaP and Ge rely on indirect

band gap transitions, less efficient for this class of materials.232

Time-refraction was observed both in Si235 and GaAs236

high-Qnanoantenna metasurfaces. Shcherbakov et al.235 re-

ported 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 shown in Fig. 15(b), this is explained by the upcon-

version 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 re-

sults achieved with combined plasmonic antenna-ENZ systems,

but comparable to what was achieved in bulk ITO222 and In:CdO

on Au.227 It is also worth noting that the system operates at

11 GW∕cm2, underlining the higher damage threshold of di-

electric antennas in comparison to plasmonic antennas.

Though high-index dielectric antennas exhibit a higher dam-

age threshold and provide more flexibility due 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 Qfactor, the stronger the potential for a modulation

as the field enhancement is higher at resonance, yet if the du-

ration of the modulating optical pulse is made shorter than

the resonance lifetime, 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.236 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

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-23 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

spectral content of the probe via spectral filtering of a supercon-

tinuum pulse.

One can also take advantage of the phase shift induced by a

high-index nanoantenna array; Guo et al.237 demonstrated nonre-

ciprocal light reflection in a Si nanobar metasurface. To this end,

a spacetime modulation was induced by the interference and beat-

ing of two pump beams with a 6-nm difference in central wave-

length and a nonreciprocal frequency shift was measured from

the reflection of the forward- and backward propagating probe

[Fig. 15(d)].

Due 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 nanostruc-

turing puts a lower cap on damage threshold and interaction

volume, these metasurfaces can rival ENZ media due 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

bandwidths, 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 en-

hance time-modulation effects, and we discuss their respective

strengths and weaknesses.

5.2.1 Quantum well polaritons

Mann et al.238 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 ab-

sorption properties. The system operates at intensities between

70 and 700 kW∕cm2, with a change in absorption on the order

of unity at the resonant frequency of the antenna. Although 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 unsatu-

rated ITO is 360 fs.211

Fig. 15 (a) Schematic of a Si metasurface engineered to support collective high-Q Fano

resonances.235 (b) Schematic of the self-induced blueshift of harmonics in the resulting Si

nanoantenna cavity:235 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

spectral evolution for an 80-fs pump pulse length in a GaAs high-Qmetasurface.236 The fringes

are measured on only one side of the spectrum due to the breaking of time-reversal symmetry

caused by the modulation. (d) Schematic of a nonreciprocal Si metasurface:237 the traveling wave

modulation breaks the space–time symmetry of the reflection phase change. Figures adapted

from Refs. 235 to 237.

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-24 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

5.2.2 Magneto-optical modulation

Although 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.239,240 A first stone was

laid when Beaurepaire et al.241 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,242 exploiting the dynamics of

the inverse Faraday effect. Particularly, Stanciu et al.243 demon-

strated 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.244 later showed helicity-dependent switch-

ing in various magnetic media, independently of the threshold

switching temperature of the medium. This opened a new path

toward ultrafast magnetic memory writing and other applica-

tions of optical switching of magnets.

Experiments involving the use of a magnetic field use a

common figure of merit, the δparameter, defined as the percent-

age change of reflection when the magnetization is reversed.

That is for a magnetic field Mand a reflection R,δ¼

½RðMÞ−Rð−MÞ∕Rð0Þ. This value is usually quite low in

bulk ferromagnetic thin films, ranging from 10−5to 10−3.

Nanostructuring can help increase this figure of merit: the cou-

pling of surface plasmon polaritons from an Au grating to a

ferromagnetic substrate allows for a stronger MOKE effect,245

leading to a δparameter of up to 24%.246 In other works, plas-

monic antennas in a magnetic field were used to control the

transmission of chiral light,247 and dielectric nanoantennas were

engineered via their electric and magnetic Mie resonances to

achieve a large Faraday rotation.248 Yet, these time-modulations

of the medium linked to the value of δdo require a switching

of the magnetization, which happens on a much slower scale

(100 ps to ns)247,249 than the optical modulation from the

MOKE itself. Hence, the ultrafast aspect of the modulation

may only come from the rotation of the polarization of light

at a given, fixed magnetization.

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.250,251 Photocurrents were generated on a timescale

of 330 fs in Huisman et al.’s work.252 As the magnetic field does

not require switching and can be maintained constant, the modu-

lation can be considered as entirely optical and ultrafast. Qiu

et al. demonstrated similar spin current generation in an antifer-

romagnetic slab in the absence of a magnetic field,253 due to sec-

ond-order optical nonlinearities. For a more detailed review of

THz emitters and wave generation, the authors recommend the

review by Feng et al.254 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 due

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.255

Artificial ENZ metamaterials exhibiting Dirac cone-like

dispersion256 or multilayered thin films make valuable candi-

dates 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∕TiO2metamaterial was demonstrated by Rashed et al.257

In parallel, enhancement of nonlinear properties around the

ENZ frequency of a multilayered Ag∕SiO2258 as well as ultrafast

modulation of absorption259 were reported. This enhancement is

explained by the dependence of the effective third-order nonlin-

ear 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 modula-

tion of a metal–insulator–metal nanocavity was observed

around the low-energy ENZ point,260 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, multilay-

ered ENZ metamaterials can be used to pin down and control

the plasmonic resonance of coupled antennas depending on the

ENZ frequency,261 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 ultrasmall mode

volumes,262 and in this way couple light efficiently to a WS2

monolayer.263 WS2and 2D materials are a promising class of

media for time-varying experiments, as we discuss in the fol-

lowing section. Although monolayers feature properties favor-

able 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∕SiO2nanoantennas.264 The efficiency of the second har-

monic was proved to originate from the multiple metal–dielec-

tric interfaces rather than symmetry-breaking, allowing for a

polarization-independent implementation of multilayered anten-

nas. Finally, magneto-optical effects have also been observed

in similar hyperbolic nanoparticles,265 opening a path toward

magneto-optical modulation 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 engi-

neering.

5.2.4 2D materials

A modern topic in nanophotonics, 2D materials exhibit many

interesting properties including high refractive index and ease

of nanostructuring,266 unconventional band structure and carrier

dynamics such as Dirac cones,267 and excitonic physics as well

as ultrahigh carrier mobility,268 resulting in strong optical non-

linearities useful for material modulation.269,270 Transition metal

dichalcogenides (TMDs), black phosphorus (BP), and graphene

feature sub-ps carrier excitation times271–273 in single layer struc-

tures. In particular, TMDs offer excitonic resonances with short

recombination times, while monolayer BP exhibits a tunable

direct bandgap from the visible to the mid-IR. In addition,

BP’s anisotropic response has also been shown to undergo

modulation under laser excitation.274 TMDs exhibit a variety

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-25 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

of useful properties for optical modulation,275 with monolayer

and low-dimensional TMDs exhibiting stronger nonlinear prop-

erties than their bulk counterparts. In Nie et al.’s work,276 a 10-fs

pulse of about 1GW∕cm2was shown to excite photocarriers in

20 fs in low-dimensional MoS2, though the change in transmis-

sion was only of 0.8%. Large second- and third-order nonlinear-

ities have also been measured in low dimensional MoS2,277,278

MoSe2,279 WS2,280,281 and WSe2,282 as well as a strong tunability

of the refractive index in WSe2,MoSe2,andMoS2.283 In addi-

tion, the strong nonlinear properties in MoSe2can be exploited

due to the comparatively high damage threshold of the mono-

layer as demonstrated by Tam et al.279 Ultimately, applications

are limited by the low-dimensionality of these materials, which

leads to a small interaction volume and short interaction time.

On the other hand, nanostructuring can again be used to enhance

the response from these systems: enhanced light–matter inter-

action has been shown in monolayers coupled with plasmonic

nanoantennas284–286 as well as with a Si waveguide.287 Nonlinear

antennas built from bulk TMDs could also provide a platform

for time-varying experiments in the same fashion as GaAs or

Si.288 Another alternative would be van der Waals heterostruc-

tures, stacked layers of TMDs that have also exhibited short rise

times of about 50 fs,289,290 with no dependence of the response

time on the twisting angle between layers.291,292 More informa-

tion on the specific spin and valley dynamics at the origin of

the ultrafast dynamics in van der Vaals heterostructures can be

found in Jin et al.’s review.293

5.3 Outlook

All-optical implementations of time-varying metasurfaces are of

great interest for the creation of new miniaturized technologies

for the control of light both in space and time. ENZ materials

such as ITO and AZO, along with high-index dielectrics, have

proven to be solid platforms for time-varying experiments and

will surely lay the ground for more complex investigations.

These nanophotonic architectures can boost both light coupling

to the nonlinear medium and field enhancement. Multilayered

ENZ metamaterials could provide new solutions to the practical

problems posed by the nature of more classical time-varying

systems, such as resonance-engineering or high permittivity

contrast. New materials and systems could pave the way for

further development of time-varying experiments, expanding

the spectral range of operation, increasing the efficiency and

the damage threshold. Quantum well polaritons and monolayer

2D materials could exhibit time-varying effects at lower ener-

gies, while magneto-optical effects would open a new frame-

work including time-varying magnetic effects, which are

ignored in current nonlinear optical experiments.

6 Conclusions

In this review article, we have presented a comprehensive over-

view of photonic time-varying media. We started by reviewing

the basic phenomenological and mathematical considerations

rooted in the behavior of Maxwell’s equations in the presence

of temporal material discontinuities, discussing the main direc-

tions of ongoing research on electromagnetic time-switching for

several applications, such as time-reversal, energy manipulation,

frequency conversion, bandwidth enhancement, and wave rout-

ing, among many others. We then continued our discussion to

consider PTCs, discussing the basic phenomenology underlying

periodic time-scattering and parametric amplification to develop

an insight that we deployed in addressing more advanced

instances of time-modulation, in particular for topological phys-

ics, non-Hermitian systems, and temporal disorder, concluding

with some brief remarks on time-modulated surfaces. We then

extended our discussion to combinations of spatial and temporal

degrees of freedom, providing an overview of the basic proper-

ties of space–time crystals, and their application to nonreciproc-

ity, as well as the engineering of synthetic motion and its

applications to optical drag and giant bianisotropic responses,

highlighting the concept of luminal amplification and spatio-

temporal localization as a new form of wave amplification

physically distinct from the conventional parametric gain, and

concluding with an overview of the wealth of opportunities and

applications for space–time metasurfaces. Finally, we reviewed

some of the most successful materials and paradigms for exper-

imental realizations of time-varying effects in the visible and IR,

indicating some of the most promising avenues recently un-

veiled for future optical experiments with time-varying media.

While the latter undoubtedly constitute the greatest long-term

challenge for this rising field of research, we believe that,

as in many other instances of scientific enquiry, the quest for

time-varying photonic systems will prove a prolific one from

both the point of view of unveiling and demonstrating funda-

mentally novel wave phenomena and for revealing new and

unexpected windows of opportunity for technological ad-

vancement.

Acknowledgments

E.G. acknowledges funding from the Engineering and Physical

Sciences Research Council via an EPSRC Doctoral Prize

Fellowship (Grant No. EP/T51780X/1) and a Junior Fellowship

of the Simons Society of Fellows (855344,EG). P.A.H. and

M.S. acknowledge funding from Fundação para a Ciência e a

Tecnologia and Instituto de Telecomunicações under project

UIDB/50008/2020. P.A.H. is funded by the CEEC Individual

program from Fundação para a Ciência e a Tecnologia with

reference CEECIND/02947/2020. R.S., J.P., and S.V. acknowl-

edge funding from the Engineering and Physical Sciences

Research Council (EP/V048880). J.P. acknowledges funding

from the Gordon and Betty More Foundation. A.A., S.Y., and

H.L. acknowledge funding from the Department of Defense,

the Simons Foundation, and the Air Force Office of Scientific

Research MURI program.

References

1. E. Knox and A. Wilson, Eds., The Routledge Companion to

Philosophy of Physics, 1st ed., Taylor and Francis Group,

p. 786, New York (2021).

2. R. Schützhold and W. G. Unruh, “Hawking radiation in an electro-

magnetic waveguide?”Phys. Rev. Lett. 95(3), 031301 (2005).

3. T. G. Philbin et al., “Fiber-optical analog of the event horizon,”

Science 319(5868), 1367–1370 (2008).

4. A. Shapere and F. Wilczek, “Classical time crystals,”Phys. Rev.

Lett. 109(16), 160402 (2012).

5. K. Sacha and J. Zakrzewski, “Time crystals: a review,”Rep.

Prog. Phys. 81(1), 016401 (2018).

6. N. Y. Yao et al., “Classical discrete time crystals,”Nat. Phys.

16(4), 438–447 (2020).

7. H. Keßler et al., “Observation of a dissipative time crystal,”Phys.

Rev. Lett. 127(4), 043602 (2021).

8. N. Engheta, “Metamaterials with high degrees of freedom: space,

time, and more,”Nanophotonics 10(1), 639–642 (2020).

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-26 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

9. L. Novotny and B. Hecht, Principles of Nano-Optics, Cambridge

University Press (2012).

10. B. E. Saleh and M. C. Teich, Fundamentals of Photonics, John

Wiley and Sons, Hoboken, New Jersey (2019).

11. R. W. Boyd, Nonlinear Optics, Academic Press, Cambridge,

Massachusetts (2020).

12. V. G. Veselago, “Electrodynamics of substances with simultane-

ously negative electrical and magnetic permeabilities,”Sov. Phys.

Usp. 10(4), 504–509 (1968).

13. J. B. Pendry, “Negative refraction makes a perfect lens,”Phys.

Rev. Lett. 85(18), 3966–3969 (2000).

14. A. Alù et al., “Epsilon-near-zero metamaterials and electromag-

netic sources: tailoring the radiation phase pattern,”Phys. Rev. B

75(15), 155410 (2007).

15. A. K. Iyer, A. Alù, and A. Epstein, “Metamaterials and metasur-

faces—historical context, recent advances, and future direc-

tions,”IEEE Trans. Antennas Propag. 68(3), 1223–1231 (2020).

16. F. R. Morgenthaler, “Velocity modulation of electromagnetic

waves,”IRE Trans. Microwave Theory Tech. 6(2), 167–172

(1958).

17. J. T. Mendonça, Theory of Photon Acceleration, CRC Press,

Boca Raton, Florida (2000).

18. J. D. Jackson, Classical Electrodynamics, John Wiley and Sons,

Hoboken, New Jersey (2012).

19. B. Auld, J. H. Collins, and H. Zapp, “Signal processing in a non-

periodically time-varying magnetoelastic medium,”Proc. IEEE

56(3), 258–272 (1968).

20. R. Fante, “Transmission of electromagnetic waves into time-

varying media,”IEEE Trans. Antennas Propag. 19(3), 417–424

(1971).

21. R. Fante, “On the propagation of electromagnetic waves through

a time-varying dielectric layer,”Appl. Sci. Res. 27(1), 341–354

(1973).

22. P. Mukherjee and S. Talwar, “Reflection and transmission of

electromagnetic waves at a moving magnetoplasma half-space,”

Radio Sci. 8(7), 653–665 (1973).

23. Y. Xiao, D. N. Maywar, and G. P. Agrawal, “Reflection and

transmission of electromagnetic waves at a temporal boundary,”

Opt. Lett. 39(3), 574–577 (2014).

24. M. A. Salem and C. Caloz, “Space-time cross-mapping and

application to wave scattering,”arXiv 1504.02012 (2015).

25. C. Caloz and Z.-L. Deck-Léger, “Spacetime metamaterials—part

II: theory and applications,”IEEE Trans. Antennas Propag.

68(3), 1583–1598 (2020).

26. M. Bakunov and A. Maslov, “Reflection and transmission of

electromagnetic waves at a temporal boundary: comment,”

Opt. Lett. 39(20), 6029 (2014).

27. D. K. Kalluri, Electromagnetics of Time Varying Complex

Media: Frequency and Polarization Transformer, CRC Press,

Boca Raton, Florida (2018).

28. A. Akbarzadeh, N. Chamanara, and C. Caloz, “Inverse prism

based on temporal discontinuity and spatial dispersion,”Opt.

Lett. 43(14), 3297–3300 (2018).

29. M. Bakunov et al., “Light scattering at a temporal boundary in

a Lorentz medium,”Opt. Lett. 46(19), 4988–4991 (2021).

30. D. M. Solís, R. Kastner, and N. Engheta, “Time-varying materi-

als in the presence of dispersion: plane-wave propagation in a

Lorentzian medium with temporal discontinuity,”Photon. Res.

9(9), 1842–1853 (2021).

31. J. Gratus et al., “Temporal boundaries in electromagnetic mate-

rials,”New J. Phys. 23(8), 083032 (2021).

32. J. Zhang, W. Donaldson, and G. Agrawal, “Temporal reflection

and refraction of optical pulses inside a dispersive medium:

an analytic approach,”J. Opt. Soc. Am. B 38(3), 997–1003

(2021).

33. J. Zhang, W. Donaldson, and G. P. Agrawal, “Time-domain

Fabry–Perot resonators formed inside a dispersive medium,”

J. Opt. Soc. Am. B 38(8), 2376–2382 (2021).

34. K. Sacha, “Modeling spontaneous breaking of time-translation

symmetry,”Phys. Rev. A 91(3), 033617 (2015).

35. D. V. Else, B. Bauer, and C. Nayak, “Floquet time crystals,”

Phys. Rev. Lett. 117(9), 090402 (2016).

36. N. Y. Yao et al., “Discrete time crystals: rigidity, criticality, and

realizations,”Phys. Rev. Lett. 118(3), 030401 (2017).

37. V. Pacheco-Peña and N. Engheta, “Temporal metamaterials with

gain and loss,”arXiv 2108.01007 (2021).

38. J. Mendonça, A. Martins, and A. Guerreiro, “Temporal beam

splitter and temporal interference,”Phys. Rev. A 68(4), 043801

(2003).

39. V. Pacheco-Peña and N. Engheta, “Antireflection temporal coat-

ings,”Optica 7(4), 323–331 (2020).

40. H. Li et al., “Temporal parity-time symmetry for extreme energy

transformations,”Phys. Rev. Lett. 127(15), 153903 (2021).

41. D. M. Pozar, Microwave Engineering, John Wiley and Sons,

Hoboken, New Jersey (2011).

42. D. Christodoulides et al., Parity-Time Symmetry and Its

Applications, Vol. 280, Springer, Singapore (2018).

43. V. Pacheco-Peña and N. Engheta, “Temporal aiming,”Light Sci.

Appl. 9(1), 129 (2020).

44. V. Pacheco-Peña and N. Engheta, “Temporal Brewster angle,”

Phys. Rev. B 104(21), 214308 (2021).

45. J. Xu, W. Mai, and D. H. Werner, “Complete polarization con-

version using anisotropic temporal slabs,”Opt. Lett. 46(6), 1373–

1376 (2021).

46. Z. Hayran, A. Chen, and F. Monticone, “Spectral causality and

the scattering of waves,”Optica 8(8), 1040–1049 (2021).

47. D. M. Sols and N. Engheta, “Functional analysis of the polari-

zation response in linear time-varying media: a generalization of

the Kramers–Kronig relations,”Phys. Rev. B 103(14), 144303

(2021).

48. E. Yablonovitch, “Spectral broadening in the light transmitted

through a rapidly growing plasma,”Phys. Rev. Lett. 31(14),

877–879 (1973).

49. E. Yablonovitch, “Self-phase modulation of light in a laser-

breakdown plasma,”Phys. Rev. Lett. 32(20), 1101–1104

(1974).

50. C.-L. Jiang, “Wave propagation and dipole radiation in a sud-

denly created plasma,”IEEE Trans. Antennas Propag. 23(1),

83–90 (1975).

51. S. Wilks, J. Dawson, and W. Mori, “Frequency up-conversion of

electromagnetic radiation with use of an overdense plasma,”

Phys. Rev. Lett. 61(3), 337–340 (1988).

52. V. Bacot et al., “Time reversal and holography with spacetime

transformations,”Nat. Phys. 12(10), 972–977 (2016).

53. G. A. Menendez and B. Maes, “Time reflection and refraction of

graphene plasmons at a temporal discontinuity,”Opt. Lett.

42(23), 5006–5009 (2017).

54. J. Wilson et al., “Temporal control of graphene plasmons,”Phys.

Rev. B 98(8), 081411 (2018).

55. A. Maslov and M. Bakunov, “Temporal scattering of a graphene

plasmon by a rapid carrier density decrease,”Optica 5(12),

1508–1515 (2018).

56. A. Shlivinski and Y. Hadad, “Beyond the Bode-Fano bound:

wideband impedance matching for short pulses using temporal

switching of transmission-line parameters,”Phys. Rev. Lett.

121(20), 204301 (2018).

57. H. Li and A. Alù, “Temporal switching to extend the bandwidth

of thin absorbers,”Optica 8(1), 24–29 (2021).

58. Y. Mazor, M. Cotrufo, and A. Alù, “Unitary excitation transfer

between coupled cavities using temporal switching,”Phys. Rev.

Lett. 127(1), 013902 (2021).

59. V. Pacheco-Peña and N. Engheta, “Spatiotemporal isotropic-to-

anisotropic meta-atoms,”New J. Phys. 23(9), 095006 (2021).

60. F. Miyamaru et al., “Ultrafast frequency-shift dynamics at

temporal boundary induced by structural-dispersion switching

of waveguides,”Phys. Rev. Lett. 127(5), 053902 (2021).

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-27 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

61. W. Mori et al., “Conversion of DC fields in a capacitor array to

radiation by a relativistic ionization front,”Phys. Rev. Lett. 74(4),

542–545 (1995).

62. M. Cirone, K. Rzaz-Dotewski, and J. Mostowski, “Photon gen-

eration by time-dependent dielectric: a soluble model,”Phys.

Rev. A 55(1), 62–66 (1997).

63. M. J. Mencagli et al., “Static-to-dynamic field conversion with

time-varying media,”arXiv:2107.11420 (2021).

64. G. Ptitcyn, M. S. Mirmoosa, and S. A. Tretyakov, “Time-modu-

lated meta-atoms,”Phys. Rev. Res. 1(2), 023014 (2019).

65. M. S. Mirmoosa et al., “Dipole polarizability of time-varying

particles,”arXiv:2002.12297 (2020).

66. G. Ptitcyn et al., “Scattering from spheres made of time-varying

and dispersive materials,”arXiv:2110.07195 (2021).

67. J. Tunesi et al., “Terahertz emission mediated by ultrafast

time-varying metasurfaces,”Phys. Rev. Res. 3(4), L042006

(2021).

68. L. Stefanini et al., “Temporal interfaces and metamaterial re-

sponse enabled by boundaries,”arXiv:2105.14350 (2021).

69. J. R. Sanmartn, “O botafumeiro: parametric pumping in the

middle ages,”Am. J. Phys. 52(10), 937–945 (1984).

70. F. Wilczek, “Quantum time crystals,”Phys. Rev. Lett. 109(16),

160401 (2012).

71. K. Sacha, Time Crystals, Springer, Cham, Switzerland (2020).

72. E. Lustig, Y. Sharabi, and M. Segev, “Topological aspects of

photonic time crystals,”Optica 5(11), 1390–1395 (2018).

73. D. Torrent, W. J. Parnell, and A. N. Norris, “Loss compensation

in time-dependent elastic metamaterials,”Phys. Rev. B 97(1),

014105 (2018).

74. D. Ramaccia et al., “Temporal multilayer structures for designing

higher-order transfer functions using time-varying metamateri-

als,”Appl. Phys. Lett. 118(10), 101901 (2021).

75. G. Castaldi et al., “Exploiting space-time duality in the synthesis

of impedance transformers via temporal metamaterials,”

Nanophotonics 10(14), 3687–3699 (2021).

76. J. G. Gaxiola-Luna and P. Halevi, “Temporal photonic (time)

crystal with a square profile of both permittivity ε(t) and per-

meability μ(t),”Phys. Rev. B 103(14), 144306 (2021).

77. L. Yuan, A. Dutt, and S. Fan, “Synthetic frequency dimensions in

dynamically modulated ring resonators,”APL Photonics 6(7),

071102 (2021).

78. L. Yuan et al., “Synthetic dimension in photonics,”Optica 5(11),

1396–1405 (2018).

79. L. Yuan, Y. Shi, and S. Fan, “Photonic gauge potential in a sys-

tem with a synthetic frequency dimension,”Opt. Lett. 41(4),

741–744 (2016).

80. A. Dutt et al., “A single photonic cavity with two independent

physical synthetic dimensions,”Science 367(6473), 59–64

(2020).

81. C. W. Peterson et al., “Strong nonreciprocity in modulated res-

onator chains through synthetic electric and magnetic fields,”

Phys. Rev. Lett. 123(6), 063901 (2019).

82. J. S. Hung et al., “Quantum simulation of the bosonic Creutz

ladder with a parametric cavity,”Phys. Rev. Lett. 127(10),

100503 (2021).

83. A. Dutt et al., “Experimental demonstration of dynamical input

isolation in nonadiabatically modulated photonic cavities,”ACS

Photonics 6(1), 162–169 (2019).

84. M. Zhang et al., “Electronically programmable photonic mol-

ecule,”Nat. Photonics 13(1), 36–40 (2019).

85. X. Ni, S. Kim, and A. Alù, “Topological insulator in two

synthetic dimensions based on an optomechanical resonator,”

Optica 8(8), 1024–1032 (2021).

86. T. T. Koutserimpas and R. Fleury, “Nonreciprocal gain in non-

Hermitian time-Floquet systems,”Phys. Rev. Lett. 120(8), 087401

(2018).

87. R. Carminati et al., “Universal statistics of waves in a random

time-varying medium,”Phys. Rev. Lett. 127(9), 4101 (2021).

88. T. T. Koutserimpas, A. Alù, and R. Fleury, “Parametric amplifi-

cation and bidirectional invisibility in PT-symmetric time-

Floquet systems,”Phys. Rev. A 97(1), 013839 (2018).

89. F. M. Arscott, “Theory and Application of Mathieu Functions.

By N. W. Mclachlan. pp. xii, 401. (Dover, New York.),”

Math. Gaz. 52(379), 94–95 (1968).

90. N. Wang, Z.-Q. Zhang, and C. T. Chan, “Photonic Floquet media

with a complex time-periodic permittivity,”Phys. Rev. B 98(8),

085142 (2018).

91. A. Bossart and R. Fleury, “Non-Hermitian time evolution: from

static to parametric instability,”Phys. Rev. A 104(4), 2225 (2021).

92. T. T. Koutserimpas and R. Fleury, “Electromagnetic fields in

a time-varying medium: exceptional points and operator sym-

metries,”IEEE Trans. Antennas Propag. 68(9), 6717–6724 (2020).

93. M. Li et al., “Topological phases and nonreciprocal edge states in

non-Hermitian Floquet insulators,”Phys. Rev. B 100(4), 045423

(2019).

94. K. Wang et al., “Topological complex-energy braiding of non-

Hermitian bands,”Nature 598(7879), 59–64 (2021).

95. S. Horsley, M. Artoni, and G. La Rocca, “Spatial Kramers–

Kronig relations and the reflection of waves,”Nat. Photonics

9(7), 436–439 (2015).

96. B. Apffel et al., “Time localization of energy in disordered time-

modulated systems,”arXiv:2102.07583 (2021).

97. Y. Sharabi, E. Lustig, and M. Segev, “Disordered photonic time

crystals,”Phys. Rev. Lett. 126(16), 163902 (2021).

98. J. Pendry, “The evolution of waves in disordered media,”J. Phys.

C Solid State Phys. 15(16), 3493–3511 (1982).

99. U. C. Fischer and D. Pohl, “Observation of single-particle plas-

mons by near-field optical microscopy,”Phys. Rev. Lett. 62(4),

458–461 (1989).

100. R. W. Wood, “XLII. On a remarkable case of uneven distribution

of light in a diffraction grating spectrum,”London Edinburgh

Dublin Philos. Mag. J. Sci. 4(21), 396–402 (1902).

101. E. Galiffi et al., “Wood anomalies and surface-wave excitation

with a time grating,”Phys. Rev. Lett. 125(12), 127403 (2020).

102. Z. Wu and A. Grbic, “Serrodyne frequency translation using

time-modulated metasurfaces,”IEEE Trans. Antennas Propag.

68(3), 1599–1606 (2020).

103. R. C. Cumming, “The serrodyne frequency translator,”Proc. IRE

45(2), 175–186 (1957).

104. X. Wang et al., “Theory and design of multifunctional space-time

metasurfaces,”Phys. Rev. Appl. 13(4), 044040 (2020).

105. D. Ramaccia et al., “Phase-induced frequency conversion and

Doppler effect with time-modulated metasurfaces,”IEEE

Trans. Antennas Propag. 68(3), 1607–1617 (2020).

106. C. Scarborough, Z. Wu, and A. Grbic, “Efficient computation of

spatially discrete traveling-wave modulated structures,”IEEE

Trans. Antennas Propag. 69(12), 8512–8525 (2021).

107. Z. Wu, C. Scarborough, and A. Grbic, “Space-time-modulated

metasurfaces with spatial discretization: free-space n-path sys-

tems,”Phys. Rev. Appl. 14(6), 064060 (2020).

108. X. Wang et al., “Space–time metasurfaces for power combining

of waves,”ACS Photonics 8(10), 3034–3041 (2021).

109. D. Ramaccia et al., “Electromagnetic isolation induced by time-

varying metasurfaces: nonreciprocal Bragg grating,”IEEE

Antennas Wirel. Propag. Lett. 19(11), 1886–1890 (2020).

110. H. B. Sedeh, M. M. Salary, and H. Mosallaei, “Time-varying

optical vortices enabled by time-modulated metasurfaces,”

Nanophotonics 9(9), 2957–2976 (2020).

111. Y. Hadad, D. L. Sounas, and A. Alù, “Space-time gradient meta-

surfaces,”Phys. Rev. B 92(10), 100304 (2015).

112. J. S. Martínez-Romero and P. Halevi, “Parametric resonances in

a temporal photonic crystal slab,”Phys. Rev. A 98(5), 053852

(2018).

113. L. Zhang et al., “A wireless communication scheme based on

space- and frequency-division multiplexing using digital meta-

surfaces,”Nat. Electron. 4(3), 218–227 (2021).

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-28 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

114. L. Zhang et al., “Space-time-coding digital metasurfaces,”Nat.

Commun. 9, 4334 (2018).

115. A. M. Shaltout et al., “Spatiotemporal light control with frequency-

gradient metasurfaces,”Science 365(6451), 374–377 (2019).

116. A. M. Shaltout, V. M. Shalaev, and M. L. Brongersma,

“Spatiotemporal light control with active metasurfaces,”Science

364(6441), eaat3100 (2019).

117. X. Wang et al., “Nonreciprocity in bianisotropic systems with uni-

form time modulation,”Phys. Rev. Lett. 125(26), 266102 (2020).

118. J. S. Martínez-Romero, O. Becerra-Fuentes, and P. Halevi,

“Temporal photonic crystals with modulations of both permittiv-

ity and permeability,”Phys. Rev. A 93(6), 063813 (2016).

119. Z.-L. Deck-Léger et al., “Uniform-velocity spacetime crystals,”

Adv. Photonics 1(5), 056002 (2019).

120. E. Galiffi, P. A. Huidobro, and J. B. Pendry, “Broadband nonre-

ciprocal amplification in luminal metamaterials,”Phys. Rev. Lett.

123(20), 206101 (2019).

121. D. L. Sounas and A. Alù, “Non-reciprocal photonics based on

time modulation,”Nat. Photonics 11(12), 774–783 (2017).

122. J. Pierce, “Theory of the beam-type traveling-wave tube,”Proc.

IRE 35(2), 111–123 (1947).

123. A. L. Cullen, “A travelling-wave parametric amplifier,”Nature

181(4605), 332 (1958).

124. A. Oliner and A. Hessel, “Wave propagation in a medium with

a progressive sinusoidal disturbance,”IRE Trans. Microwave

Theory Tech. 9(4), 337–343 (1961).

125. E. Cassedy and A. Oliner, “Dispersion relations in time-space

periodic media—part I: stable interactions,”Proc. IEEE 51(10),

1342–1359 (1963).

126. E. Cassedy, “Dispersion relations in time-space periodic media—

part II: unstable interactions,”Proc. IEEE 55(7), 1154–1168

(1967).

127. R. L. Fante, “Optical propagation in space–time-modulated me-

dia using many-space-scale perturbation theory,”J. Opt. Soc. Am.

62(9), 1052–1060 (1972).

128. J. N. Winn et al., “Interband transitions in photonic crystals,”

Phys. Rev. B 59(3), 1551–1554 (1999).

129. F. Biancalana et al., “Dynamics of light propagation in spatiotem-

poral dielectric structures,”Phys. Rev. E 75(4), 046607 (2007).

130. S. Taravati, N. Chamanara, and C. Caloz, “Nonreciprocal electro-

magnetic scattering from a periodically space-time modulated

slab and application to a quasisonic isolator,”Phys. Rev. B

96(16), 165144 (2017).

131. Y. Sivan, S. Rozenberg, and A. Halstuch, “Coupled-mode theory

for electromagnetic pulse propagation in dispersive media under-

going a spatiotemporal perturbation: exact derivation, numerical

validation, and peculiar wave mixing,”Phys. Rev. B 93(14),

144303 (2016).

132. Z. Yu and S. Fan, “Complete optical isolation created by indirect

interband photonic transitions,”Nat. Photonics 3(2), 91–94

(2009).

133. H. Lira et al., “Electrically driven nonreciprocity induced by in-

terband photonic transition on a silicon chip,”Phys. Rev. Lett.

109(3), 033901 (2012).

134. C. Caloz and Z.-L. Deck-Leger, “Spacetime metamaterials—part

I: general concepts,”IEEE Trans. Antennas Propag. 68(3),

1569–1582 (2020).

135. E. Cassedy and A. Oliner, “Dispersion relations in time-space

periodic media: part I–stable interactions,”Proc. IEEE 51(10),

1342 (1963).

136. D.-W. Wang et al., “Optical diode made from a moving photonic

crystal,”Phys. Rev. Lett. 110(9), 093901 (2013).

137. N. Chamanara et al., “Optical isolation based on space-time en-

gineered asymmetric photonic band gaps,”Phys. Rev. B 96(15),

155409 (2017).

138. D. L. Sounas, C. Caloz, and A. Alù, “Giant non-reciprocity at the

subwavelength scale using angular momentum-biased metama-

terials,”Nat. Commun. 4, 2407 (2013).

139. N. A. Estep et al., “Magnetic-free non-reciprocity and isolation

based on parametrically modulated coupled-resonator loops,”

Nat. Phys. 10(12), 923–927 (2014).

140. C. Caloz et al., “Electromagnetic nonreciprocity,”Phys. Rev.

Appl. 10(4), 047001 (2018).

141. J. V. Bladel, Relativity and Engineering, Springer Verlag, Berlin

(1984).

142. A. Fresnel, “Lettre d’augustin Fresnel à François Arago sur

l’influence du mouvement terrestre dans quelques phénomènes

d’optique,”Ann. Chim. Phys. 9,57–66 (1818).

143. H. Fizeau, “Sur les hypotheses relatives à l’ether lumineux,”

Comptes Rendus l’Acad. Sci. 33, 349 (1851).

144. J. A. Kong, Electromagnetic Wave Theory, EMW Publishing,

Cambridge, Massachusetts (2008).

145. R. Fleury et al., “Sound isolation and giant linear nonreciprocity

in a compact acoustic circulator,”Science 343(6170), 516–519

(2014).

146. S. Lannebère and M. G. Silveirinha, “Wave instabilities and uni-

directional light flow in a cavity with rotating walls,”Phys. Rev.

A94(3), 033810 (2016).

147. J. B. Pendry, “Shearing the vacuum—quantum friction,”J. Phys.

Condens. Matter. 9(47), 10301 (1997).

148. M. G. Silveirinha, “Theory of quantum friction,”New J. Phys.

16(6), 063011 (2014).

149. B. Shapiro, “Fluctuation-induced forces in the presence of mo-

bile carrier drift,”Phys. Rev. B 96(7), 075407 (2017).

150. A. I. Volokitin and B. N. J. Persson, “Quantum friction,”Phys.

Rev. Lett. 106(9), 094502 (2011).

151. Y. B. Zeldovich, “Generation of waves by a rotating body,”JETP

Lett. 14(4), 180 (1971).

152. M. G. Silveirinha, “Optical instabilities and spontaneous light

emission by polarizable moving matter,”Phys. Rev. X 4(3),

031013 (2014).

153. D. Faccio and E. M. Wright, “Nonlinear Zeldovich effect: para-

metric amplification from medium rotation,”Phys. Rev. Lett.

118(9), 093901 (2017).

154. V. K. O. Sydoruk, E. Shamonina, and L. Solymar, “Terahertz

instability of surface optical-phonon polaritons that interact with

surface plasmon polaritons in the presence of electron drift,”

Phys Plasmas 17(10), 102103 (2010).

155. T. A. Morgado and M. G. Silveirinha, “Negative Landau damping

in bilayer graphene,”Phys. Rev. Lett. 119(13), 133901 (2017).

156. B. V. Duppen et al., “Current-induced birefrigent absorption and

non-reciprocal plasmons in graphene,”2D Mater. 3(1), 015011

(2016).

157. T. A. Morgado and M. G. Silveirinha, “Drift-induced uni-

directional graphene plasmons,”ACS Photonics 5(11), 4253–

4258 (2018).

158. B. Shapiro, “Thermal fluctuations of the electric field in the pres-

ence of carrier drift,”Phys. Rev. B 82(7), 075205 (2010).

159. Y. Dong et al., “Fizeau drag in graphene plasmonics,”Nature

594(7864), 513–516 (2021).

160. W. Zhao et al., “Efficient Fizeau drag from Dirac electrons in

monolayer graphene,”Nature 594(7864), 517 (2021).

161. P. A. Huidobro et al., “Fresnel drag in space–time-modulated

metamaterials,”Proc. Natl. Acad. Sci. 116(50), 24943–24948

(2019).

162. Y. Mazor and A. Alù, “One-way hyperbolic metasurfaces based

on synthetic motion,”IEEE Trans. Antennas Propag. 68(3),

1739–1747 (2020).

163. P. A. Huidobro et al., “Homogenization theory of space-time

metamaterials,”Phys. Rev. Appl. 16(1), 014044 (2021).

164. S. T. A. Serdyukov, I. Semchenko, and A. Sihvola,

Electromagnetics of Bianisotropic Materials Theory and

Applications, 1st ed., Gordon and Breach Science Publishers,

Amsterdam (2001).

165. D. R. Smith and J. Pendry, “Homogenization of metamaterials by

field averaging,”J. Opt. Soc. Am. B 23(3), 391–403 (2006).

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-29 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

166. M. G. Silveirinha, “Metamaterial homogenization approach with

application to the characterization of microstructured composites

with negative parameters,”Phys. Rev. B 75(11), 115104 (2007).

167. C. R. Simovski and S. A. Tretyakov, “Local constitutive param-

eters of metamaterials from an effective-medium perspective,”

Phys. Rev. B 75(19), 195111 (2007).

168. A. Alù, “First-principles homogenization theory for periodic

metamaterials,”Phys. Rev. B 84(7), 075153 (2011).

169. H. T. To, “Homogenization of dynamic laminates,”J. Math.

Anal. Appl. 354(2), 518–538 (2009).

170. K. Lurie, An Introduction to the Mathematical Theory of Waves,

Springer, New York (2007).

171. D. E. Aspnes, “Local-field effects and effective-medium theory:

a microscopic perspective,”Am. J. Phys. 50(8), 704 (1982).

172. J. B. Pendry et al., “Magnetism from conductors and enhanced

nonlinear phenomena,”IEEE Trans. Microwave Theory Technol.

47(11), 2075 (1999).

173. D. N. Astrov, “Magnetoelectric effect in chromium oxide,”Sov.

Phys. JETP 13(4), 729 (1961).

174. S. Coh and D. Vanderbilt, “Canonical magnetic insulators with

isotropic magnetoelectric coupling,”Phys. Rev. B 88(12), 121106

(2013).

175. S. Taravati, “Giant linear nonreciprocity, zero reflection, and zero

band gap in equilibrated space-time-varying media,”Phys. Rev.

Appl. 9(6), 064012 (2018).

176. E. Galiffi et al., “Photon localization and Bloch symmetry break-

ing in luminal gratings,”Phys. Rev. B 104(1), 014302 (2021).

177. E. Galiffi, “New degrees of freedom for metamaterials and meta-

surfaces: from hidden spatial dimensions to dynamical struc-

tures,”PhD Thesis, Imperial College London (2020).

178. J. Pendry, E. Galiffi, and P. Huidobro, “Gain in time-dependent

media—a new mechanism,”J. Opt. Soc. Am. B 38(11), 3360–

3366 (2021).

179. J. Pendry, E. Galiffi, and P. Huidobro, “Gain mechanism in time-

dependent media,”Optica 8(5), 636–637 (2021).

180. N. Chamanara, D. G. Cooke, and C. Caloz, “Linear pulse com-

pansion based on space-time modulation,”in IEEE Int. Symp.

Antennas and Propag. and USNC-URSI Radio Sci. Meeting,

IEEE, Atlanta, GA, pp. 239–240 (2019).

181. H. B. Sedeh, M. M. Salary, and H. Mosallaei, “Topological

space-time photonic transitions in angular-momentum-biased

metasurfaces,”Adv. Opt. Mater. 8(11), 2000075 (2020).

182. Y. Hadad, J. C. Soric, and A. Alù, “Breaking temporal sym-

metries for emission and absorption,”Proc. Natl. Acad. Sci.

113(13), 3471–3475 (2016).

183. A. Shaltout, A. Kildishev, and V. Shalaev, “Time-varying meta-

surfaces and Lorentz non-reciprocity,”Opt. Mater. Express

5(11), 2459–2467 (2015).

184. L. Zhang et al., “Breaking reciprocity with space-time-coding

digital metasurfaces,”Adv. Mater. 31(41), 1904069 (2019).

185. H. Rajabalipanah et al., “Analog signal processing through

space-time digital metasurfaces,”Nanophotonics 10(6), 1753–

1764 (2021).

186. D. Oue, K. Ding, and J. Pendry, “

Cerenkov radiation in vacuum

from a superluminal grating,”arXiv:2105.13681 (2021).

187. W. J. M. Kort-Kamp, A. K. Azad, and D. A. R. Dalvit, “Space-time

quantum metasurfaces,”Phys. Rev. Lett. 127(4), 043603 (2021).

188. N. Chamanara et al., “Unusual electromagnetic modes in space-

time-modulated dispersion-engineered media,”Phys. Rev. A

97(6), 063829 (2018).

189. D. Torrent, “Strong spatial dispersion in time-modulated dielec-

tric media,”Phys. Rev. B 102(21), 214202 (2020).

190. E. Galiffi, P. A. Huidobro, and J. Pendry, “An Archimedes’screw

for light,”arXiv:2109.14460 (2021).

191. R. Fleury, A. B. Khanikaev, and A. Alù, “Floquet topological

insulators for sound,”Nat. Commun. 7, 11744 (2016).

192. L. He et al., “Floquet Chern insulators of light,”Nat. Commun.

10, 4194 (2019).

193. K. Fang, Z. Yu, and S. Fan, “Photonic Aharonov–Bohm effect

based on dynamic modulation,”Phys. Rev. Lett. 108(15), 153901

(2012).

194. Q. Lin et al., “Photonic Weyl point in a two-dimensional

resonator lattice with a synthetic frequency dimension,”Nat.

Commun. 7, 13731 (2016).

195. G. W. Milton and O. Mattei, “Field patterns: a new mathematical

object,”Proc. R. Soc. A 473(2198), 20160819 (2017).

196. O. Mattei and G. W. Milton, “Field patterns without blow up,”

New J. Phys. 19(9), 093022 (2017).

197. D. Torrent, O. Poncelet, and J.-C. Batsale, “Nonreciprocal ther-

mal material by spatiotemporal modulation,”Phys. Rev. Lett.

120(12), 125501 (2018).

198. H. Nassar et al., “Quantization of band tilting in modulated

phononic crystals,”Phys. Rev. B 97(1), 014305 (2018).

199. J. Li et al., “Transfer matrix method for the analysis of space-

time-modulated media and systems,”Phys. Rev. B 100(14),

144311 (2019).

200. M. Camacho, B. Edwards, and N. Engheta, “Achieving asymme-

try and trapping in diffusion with spatiotemporal metamaterials,”

Nat. Commun. 11, 3733 (2020).

201. Y. Wang et al., “Observation of nonreciprocal wave propagation

in a dynamic phononic lattice,”Phys. Rev. Lett. 121(19), 194301

(2018).

202. Z. Chen et al., “Efficient nonreciprocal mode transitions in

spatiotemporally modulated acoustic metamaterials,”Sci. Adv.

7(45), eabj1198 (2021).

203. X. Xu et al., “Physical observation of a robust acoustic pumping

in waveguides with dynamic boundary,”Phys. Rev. Lett. 125(25),

253901 (2020).

204. C. Cho et al., “Digitally virtualized atoms for acoustic metama-

terials,”Nat. Commun. 11, 251 (2020).

205. X. Wen et al., “Unidirectional amplification with acoustic non-

Hermitian space–time varying metamaterial,”Commun. Phys.

5,18(2022).

206. V. Bacot et al., “Phase-conjugate mirror for water waves driven

by the Faraday instability,”Proc. Natl. Acad. Sci. U. S. A.

116(18), 8809–8814 (2019).

207. K. Schultheiss et al., “Time refraction of spin waves,”Phys. Rev.

Lett. 126(13), 137201 (2021).

208. E. A. Kittlaus et al., “Non-reciprocal interband Brillouin modu-

lation,”Nat. Photonics 12(10), 613–619 (2018).

209. Z. Shen et al., “Experimental realization of optomechanically

induced non-reciprocity,”Nat. Photonics 10(10), 657–661 (2016).

210. N. Kinsey et al., “Near-zero-index materials for photonics,”Nat.

Rev. Mater. 4(12), 742–760 (2019).

211. M. Z. Alam et al., “Large optical nonlinearity of indium tin oxide in

its epsilon-near-zero region,”Science 352(6287), 795–797 (2016).

212. L. Caspani et al., “Enhanced nonlinear refractive index in ε-near-

zero materials,”Phys. Rev. Lett. 116(23), 233901 (2016).

213. T. S. Luk et al., “Enhanced third harmonic generation from the

epsilon-near-zero modes of ultrathin films,”Appl. Phys. Lett.

106(15), 151103 (2015).

214. E. G. Carnemolla et al., “Degenerate optical nonlinear enhance-

ment in epsilon-near-zero transparent conducting oxides,”Opt.

Mater. Express 8(11), 3392–3400 (2018).

215. Y. Zhou et al., “Broadband frequency translation through time

refraction in an epsilon-near-zero material,”Nat. Commun. 11,

2180 (2020).

216. V. Bruno et al., “Broad frequency shift of parametric processes

in epsilon-near-zero time-varying media,”Appl. Sci. 10(4), 1318

(2020).

217. V. Bruno et al., “Negative refraction in time-varying strongly

coupled plasmonic-antenna–epsilon-near-zero systems,”Phys.

Rev. Lett. 124(4), 043902 (2020).

218. K. Pang et al., “Adiabatic frequency conversion using a time-

varying epsilon-near-zero metasurface,”Nano Lett. 21(14),

5907–5913 (2021).

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-30 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

219. M. Ferrera and E. G. Carnemolla, “Ultra-fast transient plasmonics

using transparent conductive oxides,”J. Opt. 20(2), 024007 (2018).

220. S. Vezzoli et al., “Optical time reversal from time-dependent ep-

silon-near-zero media,”Phys. Rev. Lett. 120(4), 043902 (2018).

221. J. Bohn, T. S. Luk, and E. Hendry, “Spatiotemporal refraction of

light in an epsilon-near-zero indium tin oxide layer: frequency

shifting effects arising from interfaces,”Optica 8(12), 1532–1537

(2021).

222. C. Liu et al., “Photon acceleration using a time-varying epsilon-

near-zero metasurface,”ACS Photonics 8(3), 716–720 (2021).

223. P. Guo et al., “Large optical nonlinearity of ITO nanorods for

sub-picosecond all-optical modulation of the full-visible spec-

trum,”Nat. Commun. 7, 12892 (2016).

224. S. Vassant et al., “Berreman mode and epsilon near zero mode,”

Opt. Express 20(21), 23971 (2012).

225. J. Bohn et al., “All-optical switching of an epsilon-near-zero plas-

mon resonance in indium tin oxide,”Nat. Commun. 12, 1017

(2021).

226. M. Z. Alam et al., “Large optical nonlinearity of nanoantennas

coupled to an epsilon-near-zero material,”Nat. Photonics 12(2),

79–83 (2018).

227. Y. Yang et al., “High-harmonic generation from an epsilon-near-

zero material,”Nat. Phys. 15(10), 1022–1026 (2019).

228. C. Husko et al., “Ultrafast all-optical modulation in GaAs pho-

tonic crystal cavities,”Appl. Phys. Lett. 94(2), 021111 (2009).

229. M. R. Shcherbakov et al., “Ultrafast all-optical switching with

magnetic resonances in nonlinear dielectric nanostructures,”

Nano Lett. 15(10), 6985–6990 (2015).

230. S. Makarov et al., “Tuning of magnetic optical response in

a dielectric nanoparticle by ultrafast photoexcitation of dense

electron–hole plasma,”Nano Lett. 15(9), 6187–6192 (2015).

231. G. Della Valle et al., “Nonlinear anisotropic dielectric metasur-

faces for ultrafast nanophotonics,”ACS Photonics 4(9), 2129–

2136 (2017).

232. M. R. Shcherbakov et al., “Ultrafast all-optical tuning of direct-gap

semiconductor metasurfaces,”Nat. Commun. 8, 17 (2017).

233. G. Grinblat et al., “Efficient ultrafast all-optical modulation in a

nonlinear crystalline gallium phosphide nanodisk at the anapole

excitation,”Sci. Adv. 6(34), eabb3123 (2020).

234. R. Lemasters et al., “Deep optical switching on subpicosecond

timescales in an amorphous Ge metamaterial,”Adv. Opt. Mater.

9(15), 2100240 (2021).

235. M. R. Shcherbakov et al., “Photon acceleration and tunable

broadband harmonics generation in nonlinear time-dependent

metasurfaces,”Nat. Commun. 10, 1345 (2019).

236. N. Karl et al., “Frequency conversion in a time-variant dielectric

metasurface,”Nano Lett. 20(10), 7052–7058 (2020).

237. X. Guo et al., “Nonreciprocal metasurface with space–time phase

modulation,”Light Sci. Appl. 8(1), 123 (2019).

238. S. A. Mann et al., “Ultrafast optical switching and power limiting

in intersubband polaritonic metasurfaces,”Optica 8(5), 606–613

(2021).

239. N. Maccaferri et al., “Nanoscale magnetophotonics,”J. Appl.

Phys. 127(8), 080903 (2020).

240. A. Kirilyuk, A. V. Kimel, and T. Rasing, “Ultrafast optical

manipulation of magnetic order,”Rev. Mod. Phys. 82(3),

2731–2784 (2010).

241. E. Beaurepaire et al., “Ultrafast spin dynamics in ferromagnetic

nickel,”Phys. Rev. Lett. 76(22), 4250–4253 (1996).

242. A. V. Kimel et al., “Ultrafast non-thermal control of magnetiza-

tion by instantaneous photomagnetic pulses,”Nature 435(7042),

655–657 (2005).

243. C. D. Stanciu et al., “All-optical magnetic recording with circu-

larly polarized light,”Phys. Rev. Lett. 99(4), 047601 (2007).

244. S. Mangin et al., “Engineered materials for all-optical helicity-de-

pendent magnetic switching,”Nat. Mater. 13(3), 286–292 (2014).

245. V. I. Belotelov et al., “Enhanced magneto-optical effects in mag-

netoplasmonic crystals,”Nat. Nanotechnol. 6(6), 370–376 (2011).

246. V. I. Belotelov et al., “Plasmon-mediated magneto-optical trans-

parency,”Nat. Commun. 4, 2128 (2013).

247. I. Zubritskaya et al., “Magnetic control of the chiroptical plas-

monic surfaces,”Nano Lett. 18(1), 302–307 (2018).

248. A. Christofi et al., “Giant enhancement of faraday rotation due to

electromagnetically induced transparency in all-dielectric mag-

neto-optical metasurfaces,”Opt. Lett. 43(8), 1838–1841 (2018).

249. L. Landau and E. Lifshitz, “On the theory of the dispersion

of magnetic permeability in ferromagnetic bodies,”Phys. Z.

Sowjetunion 8,153–164 (1935).

250. T. Kampfrath et al., “Coherent terahertz control of antiferromag-

netic spin waves,”Nat. Photonics 5(1), 31–34 (2011).

251. T. Kampfrath et al., “Terahertz spin current pulses controlled by

magnetic heterostructures,”Nat. Nanotechnol. 8(4), 256–260

(2013).

252. T. J. Huisman et al., “Femtosecond control of electric currents in

metallic ferromagnetic heterostructures,”Nat. Nanotechnol.

11(5), 455–458 (2016).

253. H. Qiu et al., “Ultrafast spin current generated from an antifer-

romagnet,”Nat. Phys. 17(3), 388–394 (2021).

254. Z. Feng et al., “Spintronic terahertz emitter,”J. Appl. Phys.

129(1), 010901 (2021).

255. A. A. Orlov et al., “Controlling light with plasmonic multilayers,”

Photonics Nanostruct. Fundam. Appl. 12(3), 213–230 (2014).

256. P. Moitra et al., “Realization of an all-dielectric zero-index

optical metamaterial,”Nat. Photonics 7(10), 791–795 (2013).

257. A. R. Rashed et al., “Hot electron dynamics in ultrafast multi-

layer epsilon-near-zero metamaterials,”Phys. Rev. B 101(16),

165301 (2020).

258. S. Suresh et al., “Enhanced nonlinear optical responses of layered

epsilon-near-zero metamaterials at visible frequencies,”ACS

Photonics 8(1), 125–129 (2021).

259. J. N. Acharyya et al., “Ultrafast nonlinear pulse propagation

dynamics in metal–dielectric periodic photonic architectures,”

Adv. Mater. Interfaces 8(19), 2100757 (2021).

260. J. Kuttruff et al., “Ultrafast all-optical switching enabled by

epsilon-near-zero-tailored absorption in metal-insulator nano-

cavities,”Commun. Phys. 3(1), 114 (2020).

261. M. Habib et al., “Controlling the plasmon resonance via epsilon-

near-zero multilayer metamaterials,”Nanophotonics 9(11),

3637–3644 (2020).

262. S. R. K. C. Indukuri et al., “Ultrasmall mode volume hyperbolic

nanocavities for enhanced light–matter interaction at the nano-

scale,”ACS Nano 13(10), 11770–11780 (2019).

263. S. R. K. Indukuri et al., “WS2monolayers coupled to hyperbolic

metamaterial nanoantennas: Broad implications for light–matter-

interaction applications,”ACS Appl. Nano Mater. 3(10), 10226–

10233 (2020).

264. N. Maccaferri et al., “Enhanced nonlinear emission from single

multilayered metal–dielectric nanocavities resonating in the near-

infrared,”ACS Photonics 8(2), 512–520 (2021).

265. J. Kuttruff et al., “Magneto-optical activity in nonmagnetic hy-

perbolic nanoparticles,”Phys. Rev. Lett. 127(21), 217402 (2021).

266. R. Verre et al., “Transition metal dichalcogenide nanodisks as

high-index dielectric Mie nanoresonators,”Nat. Nanotechnol.

14(7), 679–683 (2019).

267. A. H. C. Neto et al., “The electronic properties of graphene,”Rev.

Mod. Phys. 81(1), 109–162 (2009).

268. J. R. Schaibley et al., “Valleytronics in 2d materials,”Nat. Rev.

Mater. 1(11), 16055 (2016).

269. J. You et al., “Nonlinear optical properties and applications of 2d

materials: theoretical and experimental aspects,”Nanophotonics

8(1), 63–97 (2018).

270. S. Yu et al., “2D materials for optical modulation: challenges and

opportunities,”Adv. Mater. 29(14), 1606128 (2017).

271. C. Poellmann et al., “Resonant internal quantum transitions and

femtosecond radiative decay of excitons in monolayer WSe2,”

Nat. Mater. 14(9), 889–893 (2015).

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-31 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

272. E. A. A. Pogna et al., “Photo-induced bandgap renormalization

governs the ultrafast response of single-layer MoS2,”ACS Nano

10(1), 1182–1188 (2016).

273. K. Wang et al., “Ultrafast nonlinear excitation dynamics of black

phosphorus nanosheets from visible to mid-infrared,”ACS Nano

10(7), 6923–6932 (2016).

274. S. Ge et al., “Dynamical evolution of anisotropic response in

black phosphorus under ultrafast photoexcitation,”Nano Lett.

15(7), 4650–4656 (2015).

275. A. Autere et al., “Nonlinear optics with 2d layered materials,”

Adv. Mater. 30(24), 1705963 (2018).

276. Z. Nie et al., “Ultrafast carrier thermalization and cooling dynam-

ics in few-layer MoS2,”ACS Nano 8(10), 10931–10940 (2014).

277. N. Kumar et al., “Second harmonic microscopy of monolayer

MoS2,”Phys. Rev. B 87(16), 161403 (2013).

278. A. Säynätjoki et al., “Ultra-strong nonlinear optical processes and

trigonal warping in MoS2layers,”Nat. Commun. 8, 893 (2017).

279. C. T. Le et al., “Nonlinear optical characteristics of monolayer

MoSe2,”Ann. Phys., Lpz. 528(7-8), 551–559 (2016).

280. C. Janisch et al., “Extraordinary second harmonic generation in

tungsten disulfide monolayers,”Sci. Rep. 4(1), 5530 (2014).

281. C. Torres-Torres et al., “Third order nonlinear optical response

exhibited by mono- and few-layers of WS2x,”2D Mater. 3(2),

021005 (2016).

282. J. Ribeiro-Soares et al., “Second harmonic generation in WSe2,”

2D Mater. 2(4), 045015 (2015).

283. G. Wang et al., “Tunable nonlinear refractive index of two-

dimensional MoS2,WS2, and MoSe2nanosheet dispersions

[invited],”Photonics Res. 3(2), A51–A55 (2015).

284. J. Kern et al., “Nanoantenna-enhanced light–matter interaction in

atomically thin WS2x,”ACS Photonics 2(9), 1260–1265 (2015).

285. F. Spreyer et al., “Second harmonic imaging of plasmonic pan-

charatnam-berry phase metasurfaces coupled to monolayers of

WS2x,”Nanophotonics 9(2), 351–360 (2020).

286. X. Chen et al., “Resonance coupling in hybrid gold nanohole–

monolayer WS2x nanostructures,”Appl. Mater. Today 15, 145–

152 (2019).

287. H. Chen et al., “Enhanced second-harmonic generation from two-

dimensional MoSe2on a silicon waveguide,”Light Sci. Appl.

6(10), e17060 (2017).

288. S. Busschaert et al., “Transition metal dichalcogenide resonators

for second harmonic signal enhancement,”ACS Photonics 7(9),

2482–2488 (2020).

289. X. Hong et al., “Ultrafast charge transfer in atomically thin MoS2∕

WS2heterostructures,”Nat. Nanotechnol. 9(9), 682–686 (2014).

290. F. Ceballos et al., “Ultrafast charge separation and indirect exci-

ton formation in a MoS2-MoSe2van der Waals heterostructure,”

ACS Nano 8(12), 12717–12724 (2014).

291. K. Wang et al., “Interlayer coupling in twisted WSe2∕WS2

bilayer heterostructures revealed by optical spectroscopy,”

ACS Nano 10(7), 6612–6622 (2016).

292. H. Zhu et al., “Interfacial charge transfer circumventing momen-

tum mismatch at two-dimensional van der Waals heterojunc-

tions,”Nano Lett. 17(6), 3591–3598 (2017).

293. C. Jin et al., “Ultrafast dynamics in van der Waals heterostruc-

tures,”Nat. Nanotechnol. 13(11), 994–1003 (2018).

Emanuele Galiffi is a junior fellow of the Simons Society of Fellows,

based at the Advanced Science Research Center of the City University

of New York. After receiving his PhD in November 2020, he was an

EPSRC Doctoral Prize Fellow at Imperial College London until August

2021. His research interests focus on exotic wave dynamics in time-

varying materials and metamaterials, transformation optics, plasmonics,

and polaritonics of low-symmetry materials.

Romain Tirole is a PhD candidate in Prof. Sapienza’s Complex

Nanophotonics Research Group at Imperial College London. His re-

search interests cover waves in time-varying ENZ metasurfaces as well

as dielectric nonlinear optical antennas. Before joining Imperial College,

Romain worked on thermal lensing in gold nanoparticle suspensions for

his master’s thesis at the University of Cambridge’s Nanophotonics

Centre, and earned a BSc degree studying successively at the University

of Toulouse and Imperial College London.

Shixiong Yin is a graduate assistant at the Advanced Science Research

Center of the City University of New York. He has been pursuing his PhD

in electrical engineering at City College of The City University of New York

since 2019, after receiving a bachelor’s degree in microwave engineering

from Harbin Institute of Technology, China. He is now focusing on wave

propagation in time-varying media, metamaterials, and plasmonics.

Huanan Li is a professor of physics at Nankai University. He received

his BSc degree in physics from Sichuan University, China, in 2009 and

his PhD in 2013 from National University of Singapore. Later, he studied

as a postdoctoral researcher at Wesleyan University (2014–2018) and

Photonics Initiative of CUNY Advanced Science Research Center

(2018–2021) in USA. His research interests focus on wave physics

and its methodology in non-Hermitian and time-varying systems.

Stefano Vezzoli is a research associate and a Strategic Teaching Fellow

at Imperial College London since 2018. From 2020 he is working on an

EPSRC New Horizon project grant titled 'Space-time metasurfaces for

light waves. Before joining Imperial College, he worked as a Research

Associate in Heriot-Watt University in Edinburgh, NTU in Singapore

and Pierre and Marie University in Paris. His research interests include

quantum optics, nonlinear optics, and metamaterials.

Paloma A. Huidobro received her PhD from Universidad Autónoma de

Madrid (Spain) in 2013 and in 2014 joined Imperial College as a postdoc-

toral researcher, where she held a Marie Sklodowska-Curie Fellowship.

In 2019 she took up an FCT Researcher Fellowship in Lisbon, where she

is currently a researcher at Instituto de Telecomunicações, based in

Instituto Superior Técnico–University of Lisbon. Her work is devoted to

developing theory of nano-scale light-matter interactions, nanophotonics,

and metamaterials.

Mário G. Silveirinha is a professor at the Instituto Superior Técnico–

University of Lisbon, Portugal and a Senior Researcher at Instituto de

Telecomunicações. He is an IEEE fellow, and OSA fellow, and an

APS fellow. His research interests include electromagnetism, plasmonics

and metamaterials, quantum optics, and topological effects.

Riccardo Sapienza works as professor of physics at Imperial College

London, director of the Plasmonics and Metamaterials Centre. He re-

ceived his PhD in physics joint from LENS, in the University of Florence

and ENS, Paris XI. He conducted postdoctoral studies in Spain in ICMM-

CSIC and in ICFO. His interests range from nanophotonics of complex

media, single molecule spectroscopy, and metamaterials.

Andrea Alù is the Einstein Professor and founding director at the

Photonics Initiative, Advanced Science Research Center, City University

of New York. He received his MS and PhD degrees from the University of

Roma Tre, Italy, and was the Temple Foundation Endowed Professor at

the University of Texas at Austin until 2018. His research interests focus

on metamaterials, electromagnetics, nanophotonics, and acoustics. He is

a fellow of NAI, AAAS, IEEE, Optica, APS, MRS, and SPIE.

J. B. Pendry is a condensed matter theorist and has worked at Imperial

College London since 1981. He has worked extensively on electronic and

structural properties of surfaces developing the theory of low energy dif-

fraction. He then turned his attention to photonic materials and designed

a series of metamaterials, completely novel materials with properties not

found in nature giving rise to the first material with a negative refractive

index and a prototype cloaking device.

Galiffi et al.: Photonics of time-varying media

Advanced Photonics 014002-32 Jan∕Feb 2022 •Vol. 4(1)

Downloaded From: https://www.spiedigitallibrary.org/journals/Advanced-Photonics on 18 Feb 2022

Spatio-spectral optical fission in time-varying subwavelength layers

Preprint

Full-text available

Jun 2024

Transparent conducting oxides are highly doped semiconductors that exhibit favourable characteristics when compared to metals, including reduced material losses, tuneable electronic and optical properties, and enhanced damage thresholds. Recently, the photonic community has renewed its attention towards these materials, recognizing their remarkable nonlinear optical properties in the near-infrared spectrum, a feature previously overlooked despite their long-standing application in photovoltaics and touchscreens. The exceptionally large and ultra-fast change of the refractive index, which can be optically induced in these compounds, extends beyond the boundaries of conventional perturbative analysis and makes this class of materials the closest approximation to a time-varying system, and a unique playground for studying a variety of novel phenomena within the domain of photon acceleration. Here we report the spatio-spectral fission of an ultra-fast pulse trespassing a thin film of aluminium zinc oxide with a non-stationary refractive index. By applying phase conservation to this time-varying layer, our model can account for both space and time refraction and explain in quantitative terms, the spatial separation of both the spectrum and energy. Our findings represent an example of extreme nonlinear phenomena on subwavelength propagation distances and shed light on the nature of several nonlinear effects recently reported not accounting for the full optical field distribution. Our work also provides new important insights into transparent conducting oxides transient optical properties which are critical for the ongoing research in photonic time crystals, on-chip generation of nonclassical states of light, integrated optical neural networks as well as ultra-fast beam steering and frequency division multiplexing

Observation of discrete-light temporal refraction by moving potentials with broken Galilean invariance

Article

Full-text available

Jun 2024

Refraction is a basic beam bending effect at two media’s interface. While traditional studies focus on stationary boundaries, moving boundaries or potentials could enable new laws of refractions. Meanwhile, media’s discretization plays a pivotal role in refraction owing to Galilean invariance breaking principle in discrete-wave mechanics, making refraction highly moving-speed dependent. Here, by harnessing a synthetic temporal lattice in a fiber-loop circuit, we observe discrete time refraction by a moving gauge-potential barrier. We unveil the selection rules for the potential moving speed, which can only take an integer v = 1 or fractional v = 1/q (odd q) value to guarantee a well-defined refraction. We observe reflectionless/reflective refractions for v = 1 and v = 1/3 speeds, transparent potentials with vanishing refraction/reflection, refraction of dynamic moving potential and refraction for relativistic Zitterbewegung effect. Our findings may feature applications in versatile time control and measurement for optical communications and signal processing.

Transient States to Control the Electromagnetic Response of Space-time Dispersive Media

Preprint

Full-text available

Jun 2024

In this paper, we study the dynamic formation of transients when plane waves impinge on a dispersive slab that abruptly changes its electrical properties in time. The time-varying slab alternates between air and metal-like states, whose frequency dispersion is described by the Drude model. It is shown how the physics of this complex system can be well described with the joint combination of two terms: one associated with temporal refractions and the other associated with spatial refractions. To test the validity of the approach, some analytical results are compared with a self-implemented finite-difference time-domain (FDTD) method. Results show how the transients that occurred after the abrupt temporal changes can shape the overall steady-state response of the space-time system. In fact, far from always being detrimental, these transient states can be conveniently used to perform frequency conversion or to amplify/attenuate the electromagnetic fields.

Space-time metallic metasurfaces for frequency conversion and beamforming

Article

Full-text available

Jun 2024

This paper details a class of metal-based space-time metasurfaces for application in wireless communications scenarios. Concretely, we describe space-time metasurfaces that periodically alternate their properties in time between three spatial states: “air,” “conductor,” and “grating.” We analyze the physics of these metastructures via a computationally efficient analytical technique based on the use of Floquet-Bloch series, integral equations, and circuit models. By doing so, we reveal features of these spatiotemporal metasurfaces: scattering parameters, field profiles, diffraction angles, and the nature of the space-time harmonics. The results, corroborated with a self-implemented numerical finite-difference time-domain approach, show the potential application of these space-time metasurfaces as beamformers acting in reflection, in transmission or both. The amplitude and direction of the diffracted orders can be electronically controlled with the parameters of the metasurface. Moreover, the intrinsic ability of time-modulated diffractive metasurfaces to mix and multiply frequencies is tested. We show how two different modulations can lead to the same diffraction angle but with different mixed output frequencies.

Conformal Spatiotemporal Modulation Enabled Geometric Frequency Combs

Article

Jun 2024

Non-Hermitian linear electro-optic effect in three-dimensional materials

Article

Jun 2024

Here, we present an in-depth theoretical analysis of the linear electro-optic effect in low-symmetry three-dimensional (3D) conductive materials with large Berry curvature dipoles. Our study identifies two distinct kinetic contributions to the linear electro-optic effect: a gyrotropic Hermitian (conservative) piece and a non-Hermitian term that can originate optical gain. We concentrate on the study of 3D materials belonging to the 32 (D3) point group subject to a static electric bias along the trigonal axis. Our investigation shows that doped trigonal tellurium has promising properties, with its gyrotropic electro-optic response offering the potential for realizing electrically biased electromagnetic isolators and inducing significant optical dichroism. Most notably, it is demonstrated that under sufficiently large static electric bias, tellurium's non-Hermitian electro-optic response may lead to optical gain. Using first-principles calculations, it is shown that n-doped tellurium is particularly promising, as it can host significantly larger Berry curvature dipoles than the more common p-doped tellurium.

Parametric resonances in a photonic time crystal with periodic square modulation of its permittivity ɛ ( t )

Article

Jun 2024

We study parametric resonances in a “temporal photonic crystal” or “photonic time crystal” (PTC) slab whose permittivity is assumed to be modulated periodically in time in a stepwise manner (with abrupt transitions). These resonances in the light reflected and transmitted by the PTC slab occur when the modulation frequency assumes a series of eigenvalues that are inversely proportional to the slab thickness, in addition to having a value that is twice that of the frequency of incidence. We compare the reflection and transmission spectra with those for a harmonically modulated slab and find marked, qualitative differences. These depend crucially on the modulation strength and on the impedance contrast at the slab's interfaces. In the special case of weak modulation, the differences in behavior between the two forms of modulation can be traced to a simple scaling factor. In principle, these findings could be corroborated with infrared light incident on PTC slabs tens of micrometers thick.

Title: Graphene plasmonic time crystals

Article

Full-text available

Jun 2024
PHYS STATUS SOLIDI-R

The concept of photonic crystals has recently been extended to the time domain and attracted great interest. Unfortunately, realizing photonic time crystals is a challenging task due to the practical difficulty in modulating dielectric constants with large modulation depth. This problem can be resolved using graphene, the conductivity of which is tunable with significantly large contrast. Herein, graphene plasmonic time crystals, as new kinds of photonic time crystals in atomically thin 2D material, are proposed and their optical properties are investigated. Their band structures are analytically calculated and the propagations of graphene plasmons in temporal crystalline structures are numerically evaluated. Periodically driven by temporally modulating the Fermi energy, graphene plasmons exhibit in‐gap amplification and defect‐immune topological edge states, revealing the nature as plasmonic time crystals. Graphene plasmonic time crystals are realized soon after this proposal due to the possibility of modulating its conductivity with large contrast by simple electrical gating.

Photonic Time Crystals and Parametric Amplification: Similarity and Distinction

Article

Jun 2024

Jacob B. Khurgin

Subwavelength resonant acoustic scattering in fast time-modulated media

Article

May 2024
J MATH PURE APPL

An Archimedes' screw for light

Article

Full-text available

May 2022

An Archimedes’ Screw captures water, feeding energy into it by lifting it to a higher level. We introduce the first instance of an optical Archimedes’ Screw, and demonstrate how this system is capable of capturing light, dragging it and amplifying it. We unveil new exact analytic solutions to Maxwell’s Equations for a wide family of chiral space-time media, and show their potential to achieve chirally selective amplification within widely tunable parity-time-broken phases. Our work, which may be readily implemented via pump-probe experiments with circularly polarized beams, opens a new direction in the physics of time-varying media by merging the rising field of space-time metamaterials and that of chiral systems, and offers a new playground for topological and non-Hermitian photonics, with potential applications to chiral spectroscopy and sensing. An Archimedes’ screw pumps energy into a fluid by lifting it to a higher ground. An optical screw uses helical space-time modulations in the properties of a material to grab hold of and amplify circularly polarized light of a specific handedness.

Čerenkov radiation in vacuum from a superluminal grating

Article

Full-text available

Jan 2022

Nothing can physically travel faster than light in vacuum. There are several ways proposed to bypass the light barrier and produce Čerenkov radiation (ČR) in vacuum. In this paper, we theoretically predict ČR in vacuum from a spatiotemporally modulated boundary. We consider the modulation of traveling wave type and apply a uniform electrostatic field on the boundary to generate electric dipoles. Since the induced dipoles stick to the interface, they travel at the modulation speed. When the grating travels faster than light, it emits ČR. In order to quantitatively examine this argument, we need to calculate the field scattered at the boundary. We utilize a dynamical differential method, which we developed in a previous paper, to quantitatively evaluate the field distribution in such a situation. We can confirm that all scattered fields are evanescent if the modulation speed is slower than light while some become propagating if the modulation is faster than light.

Unidirectional amplification with acoustic non-Hermitian space−time varying metamaterial

Article

Full-text available

Jan 2022

Space−time modulated metamaterials support extraordinary rich applications, such as parametric amplification, frequency conversion, and non-reciprocal transmission. The non-Hermitian space−time varying systems combining non-Hermiticity and space−time varying capability, have been proposed to realize wave control like unidirectional amplification, while its experimental realization still remains a challenge. Here, based on metamaterials with software-defined impulse responses, we experimentally demonstrate non-Hermitian space−time varying metamaterials in which the material gain and loss can be dynamically controlled and balanced in the time domain instead of spatial domain, allowing us to suppress scattering at the incident frequency and to increase the efficiency of frequency conversion at the same time. An additional modulation phase delay between different meta-atoms results in unidirectional amplification in frequency conversion. The realization of non-Hermitian space−time varying metamaterials will offer further opportunities in studying non-Hermitian topological physics in dynamic and nonreciprocal systems.

Spatiotemporal refraction of light in an epsilon-near-zero indium tin oxide layer: frequency shifting effects arising from interfaces

Article

Full-text available

Dec 2021

When light travels through a medium in which the refractive index is rapidly changing with time, the light will undergo a shift in its frequency. Significant frequency shifting effects have recently been reported for transparent conductive oxides. These observations have been interpreted as emerging from temporal changes to the propagation phase in a bulk medium resulting from temporal variations in the refractive index. It is an effect referred to as temporal refraction. Here, we show that the frequency shift in an epsilon-near-zero layer made of indium tin oxide originates not only from this bulk response but includes a significant effect resulting from temporal changes to the spatial boundary conditions. This boundary effect can lead to a dominant, opposing shift to the bulk effect for certain angles. Hence, this process gives rise to a frequency shift that can be tailored through the angle, decoupling the amplitude and phase modulation.

Terahertz emission mediated by ultrafast time-varying metasurfaces

Article

Full-text available

Oct 2021

Systems with ultrafast time-varying dielectric properties represent an emerging physical framework. We demonstrate here the observation of subcycle dynamics interacting directly with an electromagnetic source comprised of morphologically constrained photoexcited carriers in a surface nanostructure. A transition to a metallic metasurface state occurs on time scales faster than the terahertz-field period, inducing large nonlinear ultrafast phase shifts in the terahertz emission and exposing an interesting physical setting.

Temporal Parity-Time Symmetry for Extreme Energy Transformations

Article

Full-text available

Oct 2021
PHYS REV LETT

Temporal interfaces introduced by abrupt switching of the constitutive parameters of unbounded media enable unusual wave phenomena. So far, their explorations have been mostly limited to lossless media. Yet, non-Hermitian phenomena leveraging material loss and gain, and their balanced combination in parity-time (PT)-symmetric systems, have been opening new vistas in photonics. Here, we unveil the role that temporal interfaces offer in non-Hermitian physics, introducing the dual of PT symmetry for temporal boundaries. Our findings reveal unexplored interference mechanisms enabling extreme energy manipulation, and open new scenarios for time-switched metamaterials, connecting them with the broad opportunities offered by non-Hermitian phenomena.

Topological complex-energy braiding of non-Hermitian bands

Article

Full-text available

Oct 2021
NATURE

Effects connected with the mathematical theory of knots¹ emerge in many areas of science, from physics2,3 to biology⁴. Recent theoretical work discovered that the braid group characterizes the topology of non-Hermitian periodic systems⁵, where the complex band energies can braid in momentum space. However, such braids of complex-energy bands have not been realized or controlled experimentally. Here, we introduce a tight-binding lattice model that can achieve arbitrary elements in the braid group of two strands 𝔹2. We experimentally demonstrate such topological complex-energy braiding of non-Hermitian bands in a synthetic dimension6,7. Our experiments utilize frequency modes in two coupled ring resonators, one of which undergoes simultaneous phase and amplitude modulation. We observe a wide variety of two-band braiding structures that constitute representative instances of links and knots, including the unlink, the unknot, the Hopf link and the trefoil. We also show that the handedness of braids can be changed. Our results provide a direct demonstration of the braid-group characterization of non-Hermitian topology and open a pathway for designing and realizing topologically robust phases in open classical and quantum systems.

Static-to-dynamic field conversion with time-varying media

Article

Apr 2022

We theoretically demonstrate that a uniform static electric field distribution can be partially converted to radiation fields when a portion of the medium undergoes a temporal change of its permittivity. An in-depth theoretical investigation of this phenomenon is developed for a dielectric block with a steplike temporal change located inside a waveguide charged with a DC voltage source. Closed analytical expressions are derived for the radiated electric and magnetic fields. The exchange of energy between the electrostatic and electromagnetic fields is discussed. The reconciliation between the seemingly contradictory temporal and spatial boundary conditions for the electric and magnetic fields at the interface of the time-varying dielectric block is analyzed and elucidated. Our findings may provide an alternative solution for generating electromagnetic radiation based on time-varying media.

Non-Hermitian time evolution: From static to parametric instability

Article

Oct 2021

Eigenmode coalescence imparts remarkable properties to non-Hermitian time evolution, culminating in a purely non-Hermitian spectral degeneracy known as an exceptional point (EP). Here, we revisit time evolution around EPs, looking at both static and periodically modulated non-Hermitian Hamiltonians. We connect a Möbius group classification of two-level non-Hermitian Hamiltonians with the theory of Hill's equation, which unlocks a large class of analytical solutions. Together with the classification, this allows us to investigate the impact of the shape of the temporal modulation on the long-term dynamics of the system. In particular, we find that EP encircling does not predict the temporal class, and that the elaborate interplay between non-Hermitian and modulation instabilities is better understood through the lens of parametric resonance. Finally, we identify specific signatures of complex parametric resonance by exhibiting stability diagrams with features that cannot occur in traditional parametric resonance.

Scattering of light by spheres made from a time-modulated and dispersive material

Conference Paper