2.1.

Temporal Boundary Conditions and Temporal Scattering

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

Here, we assume that the medium is unbounded and homogeneous before and after a switching instance at $t=t_0$. Integrating from $t_0^{-}=t_0-ϵ$ to $t_0^{+}=t_0+ϵ$ with a $ϵ\to 0^{+}$, 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:

They ensure that the magnetic flux density $\mathbf{B}$ and the electric displacement $\mathbf{D}$ vary continuously in the time domain. Alternatively, the continuity of $\mathbf{B}$ and $\mathbf{D}$ can also be interpreted by the conservation of magnetic flux $\mathrm{\Phi }$ and electric charges $Q$, as shown by Morgenthaler¹⁶ and Auld et al.,¹⁹ respectively.

One of the most intriguing phenomena arising at a temporal discontinuity is the wave scattering dual to that occurring at spatial 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 temporal boundary conditions. Consider a monochromatic plane wave traveling in an unbounded, homogeneous, isotropic, nondispersive but time-varying medium. Both permittivity $\epsilon$ and permeability $\mu$ are assumed to abruptly switch from their initial values ${\epsilon }_1$ and ${\mu }_1$ to ${\epsilon }_2$ and ${\mu }_2$ at $t=t_0$. We can denote the incident wave with $D_x=D_1{e}^{j{\omega }_{1}t-jkz}$ and $B_y=B_1{e}^{j{\omega }_{1}t-jkz}$, where ${\omega }_1$ is the angular frequency and $k$ is the wavenumber. The field amplitudes are related by $B_1=Z_1{D}_1$ with $Z_1=\sqrt{{\mu }_1/{\epsilon }_1}$. The fields after the switching event read

$T$ and $R$ are the transmission and reflection coefficients defined for the electric displacement field, and $Z_2=\sqrt{{\mu }_2/{\epsilon }_2}$ is the new wave impedance after switching. The temporal boundary conditions [Eq. (3)] need to be satisfied everywhere in space, which implies that $k_2=k$, or equivalently

This condition ensures momentum conservation across temporal interfaces, which is expected due to the preserved spatial homogeneity.^16,17,24,25 By equating the incident and scattered $\mathbf{D}$ and $\mathbf{B}$ fields at $t=t_0$, we solve for the temporal scattering coefficients:

The coefficients for the electric field can be easily found by substituting the constitutive relation between $D$ and $E$, which are adopted more widely in the literature. It should be emphasized that the results presented in Eqs. (5) and (6) generally apply to an abrupt temporal interface under the assumptions of spatial homogeneity, isotropy, and instantaneous 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 complicated electrodynamic models of materials, such as anisotropy, dispersion, and broken translational symmetry in space, the temporal scattering is expectedly modified.^27–33 To date, research on these topics is very active, as discussed in the following sections.

2.2.

Temporal Slabs

Leveraging scattering phenomena at time interfaces, research efforts have been dedicated to the engineering of wave interference in time, for example, in the context of discrete time crystals^34–36 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 scattering from a mathematical standpoint is that space-scattering generally gives rise to boundary-value problems, whereas time-scattering leads to initial-value problems. Here, we discuss the recent progress in temporal slabs comprising one or two consecutive switching events. In Sec. 3, we discuss (quasi)periodically switched media.

The temporal interference introduced by a single temporal slab was first investigated by Mendonça et al.³⁸ Where they assume that the refractive index of a medium is switched from $n_0$ to $n_1$, and then switched back to $n_0$ after a time interval $\tau$. The total transmission and reflection amplitudes were found to be

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

More recently, it has been shown that extreme power and energy manipulation can be achieved in these types of temporal slabs once loss and gain are considered. In analogy to conventional parity-time (PT) symmetry with balanced gain and loss,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 unbounded medium is switched from a positive to a negative conductivity in time, ${\sigma }_2=-{\sigma }_3>0$, for equal duration $\mathrm{\Delta }t$ before switching back to the initial Hermitian medium, as shown in Fig. 2(d). Instantaneous material responses for both permittivity and conductivity have still been assumed here.

By introducing a general temporal scattering matrix formulation, 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 supports 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, $P_{\mathrm{tot}}(t)=P^{+}(t)+P^{-}(t)$ with $P^{\pm }(t)$ for the forward/backward propagating wave, can be significantly different from the incident one. By controlling the relative phase of two incident counterpropagating waves, one can reach drastically different total power flows for the same TPT-symmetric temporal bilayer. As an example, for $n_1=1$, $n_2=2-j0.2$, and $2{\omega }_1\mathrm{\Delta }t=3.5$, the temporal evolution of the normalized stored energy density (solid curves) and its two components $U_i=2\epsilon [{|E^{+}(t)|}^2+{|E^{-}(t)|}^2]$ [dashed curves, proportional to the total power flow in the two waves $E^{+}(t)$ and $E^{-}(t)$, and equal to the total stored energy in Hermitian media] and $U_c=2\mathrm{Re}[n(n^{*}-n)E^{+}(t)E^{-}(t{)}^{*}]$ (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 function of the relative phase of the excitations, as a dual phenomenon to laser-absorber pairs in conventional PT-symmetric systems.

2.3.

Temporal Switching in Anisotropic Media

Recent research has added new degrees of freedom to enable exotic wave transformations based on time-switching. One intriguing possibility is to consider material anisotropy. 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), in a time metamaterial. To map spatial frequencies into temporal frequencies, not only temporal invariance needs to be violated but also spatial symmetry breaking is required to bridge different momenta with different frequency channels, as shown in Fig. 3(b). The result is an inverse prism, as shown in Fig. 3(c), in which an unbounded isotropic medium with scalar refractive index $n_1$ is switched to a uniaxial medium with ${\overline{\overline{n}}}_2=\mathrm{diag}(n_{\parallel },n_{\parallel },n_{\perp })$ at $t=t_0$. The right part of Fig. 3(c) represents the isofrequency curves of the medium before and after switching. Because the medium remains homogeneous, the conservation of the wavenumber $\mathit{k}$ (momentum) still holds, leading to “birefringence” at temporal frequencies. A monochromatic wave at frequency ${\omega }_1$ propagating in the isotropic medium before switching will then be mapped to different frequencies, depending on its polarization and momentum, following:

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 frequencies. (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.

One interesting phenomenon for wave propagation in anisotropic crystals is that the group velocity does not necessarily align with the phase velocity.¹⁰ Based on this feature, the concept of temporal aiming was proposed in Ref. 43. Figure 3(d) shows an overview of this phenomenon based on isotropic-to-anisotropic switching. A $\mathit{p}$-polarized wave packet is immersed in a time-switched medium. Different switching schemes ${\overline{\overline{\epsilon }}}_r(t)$ 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 ${\theta }_1={\tan }^{-1}(k_z/k_x)$, the direction of the momentum $\mathit{k}$ after switching to the anisotropic medium remains the same ${\theta }_{2k}={\theta }_{1k}={\theta }_1$, while that of the Poynting vector $\mathbf{S}$ is redirected to

Another interesting phenomenon exploiting temporal switching 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 similarly determined by the condition that no backward wave is produced at the time interface. The temporal Brewster angle is given by the following simple expression:

Notice that the Brewster angle for $\mathrm{s}$-polarized waves is also expected if we consider biaxial anisotropy where ${\epsilon }_{2x}\ne {\epsilon }_{2y}\ne {\epsilon }_{2z}$. 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 nonlocal response (dispersion) of a material can be considered by assuming a susceptibility kernel in the form $\mathbf{D}(t)={\epsilon }_0[{\epsilon }_{\infty }\mathbf{E}(t)+\int \mathrm{d}{t}^{\prime }\chi (t,t^{\prime })\mathbf{E}(t-t^{\prime })]$, where ${\epsilon }_{\infty }$ is the background relative permittivity and $\chi (t,t^{\prime })$ is the time-dependent electric susceptibility, which must satisfy causality. Recently, the generalization of the Kramers–Kronig relations has been introduced for both adiabatic⁴⁶ and nonadiabatic⁴⁷ time-varying susceptibilities $\chi$. Research on temporal switching with material dispersion dates back to Fante’s and Felson’s work 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 $\mathbf{E}$ becomes continuous at a time interface, giving rise to additional temporal boundary conditions in addition to Eq. (3). The temporal boundary conditions in this scenario have been derived in Ref. 30 from the perspective of Parseval’s theorem and in Ref. 31 based on the balance of distributions. In the time domain, a Drude–Lorentz medium features a second-order differential equation for the electric polarization density $\mathbf{P}$:

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 demonstrating time-reversal at a temporal interface has been limited to water-wave phenomena by Fink’s group, whereby a water tank was uniformly shaken by an impulse, leading to the refocusing of the waves back to their emission point.⁵² In addition to limitations 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 exchanges 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 studied in Refs. 53 to 55 for two-dimensional (2D) graphene plasmons, as shown in Fig. 4(a). In one-dimensional (1D) transmission 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.

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.

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. unveiled how an excitation can be unitarily transferred between coupled cavities by switching the coupling strength, even if the detuning between the cavities is large.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 admittance, 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 explored by Tretyakov’s and Fleury’s groups,⁶⁴ developing a consistent 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 cavity, realized, e.g., by switching part of a structure, constitute yet another intriguing degree of freedom to be exploited. For instance, 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 photocarrier 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 spatial 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 study of more sophisticated material models, such as non-Hermitian, anisotropic, and even bianisotropic switching mechanisms, also in spatially structured systems. In the quest for extreme wave phenomena, temporal switching with exotic media has opened a brand-new avenue for electromagnetics research and beyond, although its ultimate impact will hinge upon further implementations of temporal switching experiments, some of which we will discuss in Sec. 5. In the next section, 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.

3.1.

Topology in Photonic Time-Crystals

One recent direction emerged in the context of spatial crystals is that of topologically nontrivial photonic phases and symmetry-protected edge modes found at their interface with a topologically inequivalent crystal. The natural question of whether such a framework exists for temporal crystals has recently been answered in the affirmative, although experimental observations of temporal topological edge states are still missing.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 specifically later than) a specific instant of time, at which the properties of the PTC are suddenly changed. In addition, while spatial edge states occur within the frequency gap of a material where the eigenstates of the infinite crystal are evanescent, temporal edge states in a PTC are only found within $k$-gaps, so 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. 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 potential. (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 constructed in a tight-binding model by introducing long-term coupling between the elements of a chain. In the time-domain, this can be realized by modulating the system at multiple frequencies.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.

Importantly, such a ladder does not need to be unidimensional; additional dimensions may be constructed from any degree of freedom by simply introducing additional coupling terms between a lattice of resonators, or, in a synthetic frequency dimension, between modes,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., ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_R$ and ${\mathrm{\Omega }}_N=N{\mathrm{\Omega }}_R$, thus introducing coupling between $N$’th-neighboring modes in the lattice, so the Hamiltonian above can be generalized to

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 amplification 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-modulation cannot provide loss, as a consequence of momentum conservation, whereas an imaginary component in the response parameters of a medium can be used to provide either gain or dissipation depending on its sign.

This distinction between parametric and non-Hermitian gain makes it natural to ask what more can be achieved by modulating in time the non-Hermitian component of the material response. As discussed in Sec. 2, the switching of dissipation not only impacts the amplitude of a wave but also its phase and can also generate backward waves, leading to highly nontrivial 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 ${\gamma }_1$ and ${\gamma }_2$ are coupled via a time-dependent coupling constant $k_0+\mathrm{\Delta }{k}^{\prime }\cos (\mathrm{\Omega }t)+j\mathrm{\Delta }{k}^{\prime \prime }\sin (\mathrm{\Omega }t)$, resulting in a non-Hermitian time-Floquet Hamiltonian. Normally, a periodic modulation of the system enables similar upconversion and downconversion between modes of a structure. However, it was shown that the inclusion of a non-Hermitian component in the time-modulation of the coupling constant between the resonators can hijack such mode coupling, rerouting all of the energy into one of the two resonators, thus producing nonreciprocal gain.⁸⁶

Fig. 7

(a) Two resonators coupled via a combination of constant ($k_0$), time-dependent Hermitian ($\mathrm{\Delta }{k}^{\prime }$), and time-dependent non-Hermitian ($\mathrm{\Delta }{k}^{\prime \prime }$) coupling terms. (b)–(e) Wave amplitude under excitation via port 1 in the two resonators (orange and blue) for two arbitrary values of $\mathrm{\Delta }{k}^{\prime }$ at the excitation frequency (${\omega }_1$) and the two sidebands for (b) $k_0\ne 0$ and $\mathrm{\Delta }{k}^{\prime \prime }=0$ (energy stored in both sidebands and both resonators); (c) $\mathrm{\Delta }{k}^{\prime \prime }=0$ and $k_0=0$ (energy channeled to sideband frequencies is all in the second resonator while the first only hosts the input frequency); (d) $\mathrm{\Delta }{k}^{\prime }=\mathrm{\Delta }{k}^{\prime \prime }$ and $k_0\ne 0$ (energy gained is nonreciprocally channeled into the upper sideband and it is distributed over both resonators); (e) $\mathrm{\Delta }{k}^{\prime }=\mathrm{\Delta }{k}^{\prime \prime }$ and $k_0=0$ (energy gained is nonreciprocally channeled only into the upper sideband and uniquely in the second resonator).⁸⁶ (f) A random sequence of temporal $\delta$-kicks results in (g) universal statistics observed by the energy in the system $U$. The energy at the $N$’th kick $U_N$ is plotted against the step number $N$.⁸⁷ Figures adapted from Refs. 86 and 87.

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 modulation doubles that of the incoming light.⁸⁸ Elegantly, PT 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 anisotropic transmission resonance, when two PT-symmetric time-Floquet slabs are combined together such that their respective parametric oscillations cancel out exactly. New perspectives relating exceptional point physics and parametric modulation may be also found in Refs. 90 to 92, whereas Refs. 93 and 94 investigate topological phases in non-Hermitian Floquet photonic systems.

Another interesting direction for non-Hermitian time-modulated systems relates to the temporal analog of the causality relations of conventional passive media.⁴⁶ As a well-known consequence of causality, imposed by demanding that the response function of a system is only non-zero at times following an input signal, the real and imaginary parts of the response 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-modulation must be related in order for such asymmetric frequency conversion to occur. Related concepts have also been exploited for event-cloaking, broadband absorption, and synthetic frequency dimensions.^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 conventional localization observed in spatially disordered media does not occur in a time-modulated system. Instead, in a similar fashion to the occurrence of edge states, the onset of the temporal analogue of Anderson localization manifests itself as an exponential growth in the energy of the waves. Such wave dynamics has been investigated both analytically,⁸⁷ demonstrating the universal statistics occurring in $\delta$-kicked temporal media, and numerically,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 metasurfaces (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 occupied a central role in photonics since its early days. In its interaction with waves in the far field, a 1D-grating can couple waves whose dispersion lives on the edge of the light cone by trading in-plane ($k_x$) and out-of-plane ($k_z$) momentum via both its in-plane structure and its transverse boundary, such that they must add in quadrature to the total momentum $k_0=\sqrt{\epsilon }\omega /c_0$ of 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 $k_x$ will now be conserved, whereas $k_z$ and, importantly, the wave frequency $\omega$ are now allowed to change. Taking a vertical cut of the light cone for a fixed value of $k_x$, we see that the two quantities being traded by the surface, $k_z$ and $\omega$, must not add, but rather subtract in quadrature. As a result, the locus of points described by the available modes in the far field is no longer elliptical but hyperbolic. One consequence is that an increase in frequency caused by a temporal modulation corresponds to an increase in out-of-plane momentum and vice versa, the opposite of what happens between momentum components as they interact with a spatial grating.

One direction for time-modulated surfaces is the opportunity to couple radiation to surface waves in the absence of any surface structure. Since their dispersion curve lies outside of the light cone, surface waves are typically excited by breaking translational symmetry along the surface either via a localized near-field probe 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 polaritonic materials, such as hexagonal boron nitride and ${\mathrm{MoO}}_3$, thereby circumventing the very need for surface fabrication or near-field excitation techniques.

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.

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 opportunities for advancing this known field of research both theoretically^104–112 and experimentally.^{102,113–116} Recent implementations of frequency translation with metasurfaces have been recently carried out in the microwave regime,¹⁰² and implementations with experimentally more practical discretized arrays of time- and space–time-modulated reactive elements are in steady development.^107,114 Several more experimental implementations are discussed in Sec. 5. Interestingly, the combination of bianisotropy and time-modulation has also been predicted to give rise to a nonreciprocal response (see Sec. 4.1).¹¹⁷ Specific scattering studies have also been dedicated to periodically modulated slabs,¹¹⁸ particularly in the context of parametric 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 interest 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).

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 simultaneous transmission and reception at the same frequency on the same channel without parasitic reflections, as well as other applications such as laser protection from backscattering and field enhancement. The common approach to breaking this symmetry of space is the use of gyromagnetic media, where the presence of a magnetic field breaks time-reversal symmetry. One common nonreciprocal effect encountered in basic electromagnetism is Faraday rotation, whereby the rotation of the polarization of a wave passing through a gyromagnetic medium depends on the direction from which the waves are impinging on the medium. One key drawback of gyromagnetic media, however, is their requirement of strong magnetic fields and large footprint for such effects to be appreciable, leading to bulky components, incompatible with CMOS technology. Time-modulation offers the opportunity of explicitly breaking time-reversal symmetry, thereby violating reciprocity without the use of strong magnetic fields.¹²¹

The basic idea of using ST modulation for optical isolation may be summarized in Fig. 9(g); the modulation produces transitions between two optical states that differ in both momentum $k$ and frequency $\omega$, as a result, if the system hosting the waves (e.g., a waveguide) supports multiple bands, to a pair of forward propagating modes coupled by the ST modulation may not correspond a pair of backward propagating ones. Therefore, while a forward incoming wave at frequency ${\omega }_1$ may be completely converted into a new mode with frequency ${\omega }_2$, a backward wave at ${\omega }_1$ will not be converted as it will be mismatched in frequency and/or momentum. Hence, by introducing a narrowband filter at ${\omega }_2$, the system allows propagation only in the backward direction, thus achieving optical isolation,^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 asymmetric bandgap in the photonic dispersion of a material, allowing propagation along one direction only.¹³⁷

Ring-resonators constitute perhaps the most promising components for electromagnetic nonreciprocity. The idea of using ring resonators for nonreciprocity makes use of the concept of angular momentum biasing [Fig. 10(a)]; a circular, directional bias splits the degeneracy between clockwise and anticlockwise states, effectively mimicking the Zeeman splitting of atomic physics [Fig. 10(b)]. In practice, this can be achieved with, e.g., three elements periodically modulated with a 120-deg phase difference between them [Fig. 10(c)]. Let us consider the two-port system formed by two linear waveguides coupled to the biased ring resonator shown in Fig. 10(d). Due to the splitting between clockwise and anticlockwise states, if the channel waveguide (on the right of the ring in the panel) is excited at a frequency resonant with the anticlockwise mode from the bottom port, its power will be mainly rerouted by the ring resonator into the drop channel (left of the ring). On the contrary, a wave impinging from the top of the waveguide, which would normally couple to the clockwise state, would now be off-resonance, thus coupling poorly to the mode of the resonator and being largely transmitted to the bottom port of the channel waveguide.¹³⁸ This strategy can be used to realize electromagnetic 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 impeding transmission in the opposite sense (i.e., $2\to 1$, $3\to 2$, and $1\to 3$). Figure 10(f) shows the transmission from 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 electromagnetic 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.

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.

Multiple extensive reviews on space–time modulation focusing on nonreciprocity have recently been published, so we refer the reader to them for further details and move on to illustrate further opportunities for space–time media.^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 electromagnetic 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 amplification^{146,151–153} among others.³

Unfortunately, it is impractical to take technological advantage of many of the features highlighted in the previous paragraph, because a realistic value for the velocity of a moving body is many orders of magnitude smaller than the speed of light. Interestingly, the actual physical motion of a macroscopic body may be imitated by a drift current in a solid state material with high mobility .^{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 response 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 characterized, in the dispersionless limit, by the following constitutive relations:

Fig. 11

(a)–(c) A synthetic space–time-modulated crystal (a) is formally equivalent to a fictitious moving bianisotropic time-invariant crystal (b) when $v_m<c_0$. The velocity of the equivalent moving crystal is identical to the modulation speed ${\mathbf{v}}_{\mathbf{m}}$; in the long-wavelength limit, the system response can be homogenized (c) and the crystal behaves as a uniaxial dielectric moving with speed ${\mathbf{v}}_{\mathbf{D}}$. The sign of $v_D$ is 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 spatiotemporally modulated, the dispersion is symmetric, and the equivalent drag velocity vanishes (blue lines). However, when both $\epsilon$ and $\mu$ are modulated, the dispersion is asymmetric (red lines, assuming $v_D>0$). The waves that propagate in the direction of ${\mathbf{v}}_{\mathbf{D}}$ have a larger group velocity than the waves that propagate in the opposite direction.¹⁶¹ (e) Drag velocity $v_D$ and (f) effective bianisotropic coefficient ${\xi }_{\mathrm{ef}}$ as a function of the modulation speed $v_m$ for 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.

In the above, ${\mathbf{1}}_t=\widehat{\mathbf{y}}\otimes \widehat{\mathbf{y}}+\widehat{\mathbf{z}}\otimes \widehat{\mathbf{z}}$ and $\otimes$ represents the tensor product of two vectors. As seen, the coordinate transformation originates a bianisotropic-type coupling determined by the magnetoelectric tensors ${\widehat{\xi }}^{\prime }$ and ${\widehat{\zeta }}^{\prime }$,¹⁶⁴ such that the electric displacement vector ${\mathbf{D}}^{\prime }$ and the magnetic induction ${\mathbf{B}}^{\prime }$ depend on both the electric and magnetic fields ${\mathbf{E}}^{\prime }$ and ${\mathbf{H}}^{\prime }$. Indeed, as further discussed below, one of the peculiar features of the traveling-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 difficulties, but it is relevant to mention that it is also possible to obtain a time-invariant response with a Lorentz coordinate transformation. Such a solution is restricted to subluminal modulation velocities, and thereby the Galilean transformation is typically the preferred one.

Interestingly, the constitutive relations [Eq. (22)] in the new coordinate system are reminiscent of those of a moving dielectric 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 $v_m$ is 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 precisely linked to that of a fictitious moving medium in the subluminal case. In fact, as previously noted, a suitable Lorentz transformation makes the response [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 wavelength 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 modeling of the relevant phenomena but also because it clearly pinpoints 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 describes the dynamics of the envelopes of the electromagnetic fields.¹⁸ In particular, in the traditional framework, there is no time averaging. Due to this reason, standard homogenization approaches are not directly applicable to spacetime crystals, where the microscopic response of the system depends simultaneously on space and on time. At present, there is no general effective medium theory for spacetime crystals. Fortunately, the particular class of spacetime crystals with traveling wave modulations 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 working 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 $\epsilon$ and permeability $\mu$ are independent of the $y$ and $z$ coordinates. It is well known that, for stratified systems, the components of the $\mathbf{E}$ and $\mathbf{H}$ fields parallel to the interfaces ($y$ and $z$ components) and the components of the $\mathbf{D}$ and $\mathbf{B}$ fields normal to the interfaces ($x$ components) are constant 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 $⟨{\mathbf{D}}^{\prime }⟩$ and $⟨{\mathbf{B}}^{\prime }⟩$ to the spatially averaged $⟨{\mathbf{E}}^{\prime }⟩$ and $⟨{\mathbf{H}}^{\prime }⟩$.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:

Remarkably, when both $\epsilon$ and $\mu$ are modulated in spacetime, ${\xi }_{\mathrm{ef}}$ can be nontrivial. In other words, notwithstanding that at the microscopic level there is no magnetoelectric coupling [Eq. (20)], the effective medium is characterized by a bianisotropic response in the static limit. While it is not unusual that the complex wave interactions in a metamaterial without inversion symmetry can result in a magnetoelectric coupling,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 characterized 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 conventional metamaterial are necessarily decoupled in the static limit and there is no bianisotropy. In contrast, for a space–time-modulated system, the electric and magnetic fields are never fully decoupled, as the time modulation of the material implies always some nontrivial time dynamics. The strength and sign of ${\xi }_{\mathrm{ef}}$ can be tuned by changing the modulation speed and the profiles of the permittivity and permeability. As shown in Fig. 11(f), ${\xi }_{\mathrm{ef}}$ can be exceptionally large for modulation speeds approaching the speed of light in the relevant materials. This results in giant nonreciprocity, as has also been shown through Floquet–Bloch expansions.¹⁷⁵ The transition from the subluminal to the superluminal regime is marked by a change in sign of ${\xi }_{\mathrm{ef}}$. Furthermore, it can be shown that constitutive relations [Eq. (26)] are identical to those of a fictitious moving anisotropic dielectric characterized in the respective co-moving frame by the permittivity ${\widehat{\epsilon }}_{\mathit{D}}$ and by the permeability ${\widehat{\mu }}_{\mathit{D}}$. The equivalent medium moves with a speed $v_D$ with respect to the lab frame. The parameters ${\widehat{\epsilon }}_{\mathrm{D}}$, ${\widehat{\mu }}_{\mathrm{D}}$, and $v_D$ are determined univocally by ${\widehat{\epsilon }}_{\mathrm{ef}}$, ${\widehat{\mu }}_{\mathrm{ef}}$, and ${\xi }_{\mathrm{ef}}$.¹⁶³ In general, $v_D$ may be rather different from the modulation speed $v_m$, both in amplitude and sign. The velocity $v_D$ can be nonzero only if $\epsilon$ and $\mu$ are both modulated in spacetime, i.e., only if ${\xi }_{\mathrm{ef}}\ne 0$. 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 $c_0$ [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 $\mathrm{sgn}(v_{\mathit{D}})\widehat{\mathbf{x}}$, moves faster than a wave propagating in the opposite direction, due to the synthetic Fresnel-drag effect.¹⁴³ A clear fingerprint of the Fresnel drag can be detected in the dispersion of $\omega$ versus $k$ of the electromagnetic modes of the spacetime crystal. In fact, when both $\epsilon$ and $\mu$ are modulated in time, the slopes of the dispersion of the photonic states near $k=0$ depend 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: $\epsilon /\mu =\mathrm{const.}$. In fact, the dispersion of a matched photonic crystal is generally linear for all frequencies. However, as we show in the next section, even this statement can be violated in an exotic class of spacetime media recently termed luminal metamaterials.

4.3.

Luminal Amplification and Spatiotemporal Localization

If the speed of a space–time modulation falls within the modulation velocity regime in Eq. (19) (the extent of this luminal regime increases with the modulation amplitude), the relevant wave physics changes dramatically, entering a very peculiar unstable phase whose underlying mechanism is, however, completely distinct from the parametric amplification processes described in Secs. 2 and 3.¹²⁰ In this regime, all forward bands become almost degenerate, as the frequency-momentum reciprocal lattice vectors effectively align with the forward bands. This regime marks a transition between the subluminal and superluminal regimes in Figs. 9(d) and 9(e), respectively. Due to the resulting strong degeneracy between forward-traveling 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.

Fig. 12

(a) Time dependence of the electric field intensity at the output port transmitted through different thicknesses (blue to red) $d$ of luminal crystal. (b) In the gain region, the permittivity gradient is positive, and vice versa. The gain is maximum at the points (in time or space) where the velocity of the waves matches the velocity of the grating. (c), (d) Power spectra of the transmitted waves (c) for input frequency ${\omega }_0=8\mathrm{\Omega }$, with $\mathrm{\Omega }$ the modulation frequency, and (d) for a DC input field (${\omega }_0=0$), showing the frequency generation responsible for the overall gain.^120,177 (e), (f) Trajectories of a point of constant phase along the spacetime variable $X=x-v_{m}t$: (e) outside of the localization regime (resulting in oscillations 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.

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 $U$:

4.4.

Space–Time Metasurfaces

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

Systems such as isolators [Fig. 13(a)], nonreciprocal phase shifters [Fig. 13(b)], and circulators [Fig. 13(c)] using space–time-modulated metasurfaces were designed, extending magnet-free nonreciprocity to surface implementations.^{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 voltage-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, opportunities for nonreciprocal hyperbolic propagation,¹⁶² as well as vacuum Čerenkov radiation¹⁸⁶ using space–time metasurfaces have recently been highlighted.

Fig. 13

Examples of applications of space–time metasurfaces. (a) An isolator based on the unidirectional excitation of evanescent modes. (b) Waves scattering off an STM experience a nonreciprocal 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 ${\omega }_0$ coming 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 azimuthal section is temporally modulated with a different relative phase, imparting the necessary angular-momentum bias. Figures adapted from Refs. 104, 108, 114, and 181.

Another application of space–time metasurfaces that was recently proposed is that of power combining of waves, a long-standing challenge in laser technology. Here, the strategy consists of using a spatiotemporal modulation of a surface to effectively trade the difference in angle of incidence between a discrete set of incoming waves with identical frequency in exchange for a difference between the (equally spaced) frequencies of the outgoing waves, which emerge at 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 several 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 electromagnetic media, mimicking the Archimedean screw for fluids, have recently been proposed, and may be realized with circularly polarized pump-probe experiments.¹⁹⁰ Topology also occupies 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 synthetic frequency dimension.¹⁹⁴ The idea of space–time media has also been formalized in the interesting mathematical formulation of “field patterns.” These objects arise from the consecutive spatial and temporal scattering in a material that features a checkerboard-like structure of its response parameters in space and time and may offer new angles to study space–time metamaterials in the future.^170,195,196 Finally, besides photonic systems, nonelectromagnetic dynamical metamaterials have already acquired significant momentum, with several theoretical proposals^197–199 and experimental realizations, including asymmetric charge diffusion,²⁰⁰ nonreciprocity,^145,201,202 Floquet topological insulators¹⁹¹ and topological pumping²⁰³ among many others. In particular, digitally controlled acoustic meta-atoms have recently attracted significant interest to probe time-modulation physics in acoustics.^204,205

Having discussed the wealth of potential new physics enabled 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.

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, modulation effects originated by short laser pulses interacting with a nonlinear medium occur on sub-ps timescales, well beyond the reach of electro-optic modulation, and very close to the time of an optical cycle (1 to 10 fs regime). Optical modulation of a medium stems from a change in $\epsilon$, the relative permittivity, due to the nonlinear response in a material. In semiconductors, these effects are often driven by out-of-equilibrium electronic populations, called hot-electrons. Nonlinear effects in semiconductors arise because of electron excitation of either real or imaginary states. Transitions through real states are realized by photocarrier excitation, that is intraband or interband transitions of the electrons within the active medium. The redistribution of electrons within the valence and/or conduction bands will affect the permittivity and thus the refractive index of the material on timescales limited either by the rate of transfer of energy from the light beam to the electrons or by the light pulse duration. This allows for sub-ps modulation of $\epsilon$ in a variety of media of interest for photonics, including dielectrics and metals. Transitions through virtual states are, on the other hand, much faster but lack the strength of photocarrier excitations: nonlinear processes via virtual states, such as four-wave-mixing and sum-frequency-generation, usually exhibit low efficiencies. Both photocarrier excitation and nonlinear optical modulation induce ultrafast modulation of $\epsilon$, 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-technology compatible TCOs with very strong nonlinear optical response,^211–214 which makes them good candidates as platforms for time-varying metasurfaces. Particularly, intraband transitions in ITO lead to a strong redistribution of the effective mass of conducting electrons and thus an efficient shifting of the plasma frequency in the material’s Drude dispersion as shown in Fig. 14(a). As shown by Alam et al. in 2016,²¹¹ 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 $\mathrm{\Delta }n/n$, where $n$ is the refractive index and $\mathrm{\Delta }n$ is 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 $\mathrm{\Delta }n$.

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 $770{\mathrm{GW}/\mathrm{cm}}^2$ in a degenerate pump-probe experiment at 1400 nm.²¹⁶ (g) Strong coupling between a plasmonic antenna and an ENZ thin film:²¹⁷ electric field $|E{|}^2$ distribution obtained from FDTD for an ITO layer thickness of 40 nm and Au antenna length of 400 nm. (h) Resulting field distribution as a function of 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 $2{\mathrm{GW}/\mathrm{cm}}^2$) leads to a shift near zero delay. Figures adapted from Refs. 216 to 218.

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 metasurface 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 significantly smaller portions of space. In this work, it was shown that, as expected, a shorter pulse will lead to a stronger shift in frequency due to a larger $\mathrm{d}n/\mathrm{d}t$ during the short propagation time within the ITO slab, but more interestingly, that saturation happens at lower energies for longer pulses. This indicates that 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 increase the optical field enhancement and lead to efficient ultrafast modulation as demonstrated by Guo et al.’s work²²³ with ITO nanorods. In this work, the thickness of the medium is $2.6\mu \mathrm{m}$, a length sufficient to cause significant loss upon propagation. An alternative to nanostructuring the ITO to increase the field enhancement consists of exploiting the plasmonic mode exhibited by flat ENZ thin films.²²⁴ In Bohn et al.’s work,²²⁵ the plasmonic ENZ mode was excited using a Kretschmann configuration, leading to a change in reflectance of 45%.

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

5.1.3.

High-index dielectric metasurfaces as time-varying media

In the quest for resonantly enhanced materials for time-modulation, high-index dielectrics provide an alternative to ENZ structures, as they offer possibilities for various resonant architectures and a high damage threshold. In particular, nanoantennas with Mie resonances or bound states in the continuum allow for strong time-modulation by combining efficient coupling to the active medium and a good field enhancement. That is because even though $\mathrm{\Delta }n/n$ is low in a high-index dielectric, the change of phase of the metasurface can be strong when the material is modulated near a resonance. Shifts in the resonance of a photonic crystal²²⁸ were demonstrated, as well as in Mie-resonant systems, in a range of materials ranging from Si^229–231 to GaAs,²³² GaP,²³³ and Ge.²³⁴ GaAs exhibits stronger modulation efficiencies as it features a direct bandgap in the visible 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-$Q$ nanoantenna metasurfaces. Shcherbakov et al.²³⁵ reported a 8.3-THz shift in the third harmonic signal from a Si metasurface with a collective Fano resonance as shown in Fig. 15(a), i.e., around 2 THz shift in the self-modulated pump signal. As shown in Fig. 15(b), this is explained by the upconversion to third harmonic of the time-refracted pump field, with the change of index of the nanoantenna cavity originating from the pump itself as well. This is a smaller shift than the best results achieved with combined plasmonic antenna-ENZ systems, but comparable to what was achieved in bulk ITO²²² and In:CdO on Au.²²⁷ It is also worth noting that the system operates at $11{\mathrm{GW}/\mathrm{cm}}^2$, underlining the higher damage threshold of dielectric antennas in comparison to plasmonic antennas.

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-$Q$ metasurface.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.

Though high-index dielectric antennas exhibit a higher damage 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 $Q$ factor, the stronger the potential for a modulation as the field enhancement is higher at resonance, yet if the duration of the modulating optical pulse is made shorter than the resonance lifetime, 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 spectral content of the probe via spectral filtering of a supercontinuum pulse.

One can also take advantage of the phase shift induced by a high-index nanoantenna array; Guo et al.²³⁷ demonstrated nonreciprocal light reflection in a Si nanobar metasurface. To this end, a spacetime modulation was induced by the interference and beating of two pump beams with a 6-nm difference in central wavelength and a nonreciprocal frequency shift was measured from the reflection of the forward- and backward propagating probe [Fig. 15(d)].

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 nanostructuring 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 enhance time-modulation effects, and we discuss their respective strengths and weaknesses.

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 energies, while magneto-optical effects would open a new framework including time-varying magnetic effects, which are ignored in current nonlinear optical experiments.